Ergodic Properties and KMS Conditions

on C^{∗}-Symbolic Dynamical Systems

Kengo Matsumoto

Received: March 3, 2010 Revised: March 12, 2011

Communicated by Joachim Cuntz

Abstract. A C^{∗}-symbolic dynamical system (A, ρ, Σ) consists of a unital C^{∗}-algebra A and a finite family {ρ_{α}}_{α∈Σ} of endomorphisms ρ_{α} of A indexed by symbols α of Σ satisfying some conditions. The endomorphisms ρ_{α}, α ∈ Σ yield both a subshift Λ_{ρ} and a C^{∗}-algebra O_{ρ}. We will study ergodic properties of the positive operator λ_{ρ} = ∑_{α∈Σ} ρ_{α} on A. We will next introduce KMS conditions for continuous linear functionals on O_{ρ} under gauge action at inverse temperature taking its value in complex numbers. We will study relationships among the eigenvectors of λ_{ρ} in A^{∗}, the continuous linear functionals on O_{ρ} satisfying KMS conditions and the invariant measures on the associated one-sided shifts. We will finally present several examples of continuous linear functionals satisfying KMS conditions.

2000 Mathematics Subject Classification: Primary 46L55, 37B10; Secondary 37D35. Keywords and Phrases: C^{∗}-algebra, symbolic dynamics, subshift, ergodic, KMS condition, invariant measure.

1. Introduction

D. Olesen and G. K. Pedersen [37] have shown that the C^{∗}-dynamical system (O_{N}, α, R) for the Cuntz algebra O_{N} with gauge action α admits a KMS state at the inverse temperature γ if and only if γ = logN, and the admitted KMS state is unique. By Enomoto-Fujii-Watatani [9], the result has been generalized to the Cuntz-Krieger algebras O_{A} as γ = logr_{A}, where r_{A} is the Perron-Frobenius eigenvalue for the irreducible matrix A with entries in {0, 1}. These results are generalized to several classes of C^{∗}-algebras having gauge actions (cf. [7], [10], [11], [15], [17], [18], [27], [35], [36], [41], etc.).

Cuntz-Krieger algebras are considered to be constructed by finite directed graphs which yield an important class of symbolic dynamics called shifts of finite type. In [29], the author has generalized the notion of finite directed graphs to a notion of labeled Bratteli diagrams having shift like maps, which we call λ-graph systems. A λ-graph system 𝔏 gives rise to both a subshift Λ_{𝔏} and a C^{∗}-algebra O_{𝔏} with gauge action. Some topological conjugacy invariants of subshifts have been studied through the C^{∗}-algebras constructed from λ-graph systems ([30]). A C^{∗}-symbolic dynamical system is a generalization of both a λ-graph system and an automorphism of a unital C^{∗}-algebra ([31]). It is a finite family {ρ_{α}}_{α∈Σ} of endomorphisms indexed by a finite set Σ of a unital C^{∗}-algebra A such that ρ_{α}(Z_{A}) ⊂ Z_{A} for α ∈ Σ and ∑_{α∈Σ} ρ_{α}(1) ≥ 1 where Z_{A} denotes the center of A. A finite directed labeled graph G gives rise to a C^{∗}-symbolic dynamical system (A_{G}, ρ^{G}, Σ) such that A_{G} = C^{N} for some N ∈ N. A λ-graph system 𝔏 also gives rise to a C^{∗}-symbolic dynamical system (A_{𝔏}, ρ^{𝔏}, Σ) such that A_{𝔏} is C(Ω_{𝔏}) for some compact Hausdorff space Ω_{𝔏} with dim Ω_{𝔏} = 0. A C^{∗}-symbolic dynamical system (A, ρ, Σ) yields a subshift denoted by Λ_{ρ} over Σ and a Hilbert C^{∗}- bimodule (ϕ_{ρ}, H_{A}^{ρ}) over A. By using general construction of C^{∗}-algebras from Hilbert C^{∗}-bimodules established by M. Pimsner [40], a C^{∗}-algebra denoted by O_{ρ} from (ϕ_{ρ}, H_{A}^{ρ}) has been introduced in [31]. The C^{∗}-algebra O_{ρ} is realized as the universal C^{∗}-algebra generated by partial isometries S_{α}, α ∈ Σ and x ∈ A subject to the relations:

∑_{γ∈Σ} S_{γ}S_{γ}^{∗} = 1,  S_{α}S_{α}^{∗}x = xS_{α}S_{α}^{∗},  S_{α}^{∗}xS_{α} = ρ_{α}(x)

for all x ∈ A and α ∈ Σ. We call the algebra O_{ρ} the C^{∗}-symbolic crossed product of A by the subshift Λ_{ρ}. The gauge action on O_{ρ} denoted by ρ̂ is defined by

ρ̂_{z}(x) = x, x ∈ A and ρ̂_{z}(S_{α}) = zS_{α}, α ∈ Σ

for z ∈ C, |z| = 1. If A = C(X) with dim X = 0, there exists a λ-graph system 𝔏 such that Λ_{ρ} is the subshift presented by 𝔏 and O_{ρ} is the C^{∗}-algebra O_{𝔏} associated with 𝔏. If in particular, A = C^{N}, the subshift Λ_{ρ} is a sofic shift and O_{ρ} is a Cuntz-Krieger algebra. If Σ = {α} an automorphism α of a unital C^{∗}-algebra A, the C^{∗}-algebra O_{ρ} is the ordinary C^{∗}-crossed product A ×_{α} Z. Throughout the paper, we will assume that the C^{∗}-algebra A is commutative. For a C^{∗}-symbolic dynamical system (A, ρ, Σ), define the positive operator λ_{ρ} on A by

λ_{ρ}(x) = ∑_{α∈Σ} ρ_{α}(x), x ∈ A.

We set for a complex number β ∈ C the eigenvector space of λ_{ρ}

E_{β}(ρ) = {φ ∈ A^{∗} | φ ∘ λ_{ρ} = βφ}.  (1.1)

Let Sp(ρ) be the set of eigenvalues of λ_{ρ} defined by

Sp(ρ) = {β ∈ C | E_{β}(ρ) ≠ {0}}.  (1.2)

Let r_{ρ} denote the spectral radius of λ_{ρ} on A. We set T_{ρ} = \frac{1}{r_{ρ}} λ_{ρ}. (A, ρ, Σ) is said to be power-bounded if the sequence ∥T_{ρ}^{k}∥, k ∈ N is bounded. A state φ on A is said to be invariant if φ ∘ T_{ρ} = φ. If an invariant state is unique, (A, ρ, Σ) is said to be uniquely ergodic. If lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} T_{ρ}^{k}(a) exists in A for a ∈ A, (A, ρ, Σ) is said to be mean ergodic. If there exists no nontrivial ideal of A invariant under λ_{ρ}, (A, ρ, Σ) is said to be irreducible. It will be proved that a mean ergodic and irreducible (A, ρ, Σ) is uniquely ergodic and power-bounded (Theorem 3.12). Let A = [A(i, j)]_{i,j=1}^{N} be an irreducible matrix with entries in {0, 1}, and S_{i}, i = 1, …, N be the canonical generating family of partial isometries of the Cuntz-Krieger algebra O_{A}. Let A_{A} be the C^{∗}-subalgebra of O_{A} generated by the projections S_{j}S_{j}^{∗}, j = 1, …, N. Put Σ = {1, …, N} and ρ_{i}^{A}(x) = S_{i}^{∗}xS_{i}, x ∈ A_{A}, i ∈ Σ. Then the triplet (A_{A}, ρ^{A}, Σ) yields an example of C^{∗}-symbolic dynamical system such that its C^{∗}-symbolic crossed product O_{ρ^{A}} is the Cuntz-Krieger algebra O_{A}. The above space E_{β}(ρ) is identified with the eigenvector space of the matrix A for an eigenvalue β. By Enomoto-Fujii-Watatani [9], a tracial state φ ∈ E_{β}(ρ^{A}) on A_{A} extends to a KMS state for gauge action on O_{A} if and only if β = r_{A} the Perron-Frobenius eigenvalue, and its inverse temperature is log r_{A}. The admitted KMS state is unique. In this paper, we will study the space E_{β}(ρ) of a general C^{∗}-symbolic dynamical system (A, ρ, Σ) for a general eigenvalue β in C not necessarily maximum eigenvalue and then introduce KMS condition for inverse temperature taking its value in complex numbers. In this generalization, we will study possibility of extension of a continuous linear functional on A belonging to the eigenvector space E_{β}(ρ) to the whole algebra O_{ρ} as a continuous linear functional satisfying KMS condition. For a C^{∗}-algebra with a continuous action of the one-dimensional torus group T = R/2πZ and a complex number β ∈ C, we will introduce KMS condition for a continuous linear functional without assuming its positivity at inverse temperature Logβ. Let B be a C^{∗}-algebra and α : T ⟶ Aut(B) be a continuous action of T to the automorphism group Aut(B). We write a complex number β with |β| > 1 as β = re^{iθ} where r > 1, θ ∈ R. Denote by B^{∗} the Banach space of all complex valued continuous linear functionals on B. Definition. A continuous linear functional φ ∈ B^{∗} is said to satisfy KMS condition at Logβ if φ satisfies the condition

φ(yα_{i log r}(x)) = φ(α_{θ}(x)y),  x ∈ B^{a}, y ∈ B,  (1.3)

where B^{a} is the set of analytic elements of the action α : T ⟶ Aut(B) (cf.[3]). We will prove

Theorem 1.1. Let (A, ρ, Σ) be an irreducible and power-bounded C^{∗}-symbolic dynamical system. Let β ∈ C be a complex number with |β| > 1.

(i) If β ∈ Sp(ρ) and |β| = r_{ρ} the spectral radius of the positive operator λ_{ρ} : A ⟶ A, then there exists a nonzero continuous linear functional on O_{ρ} satisfying KMS condition at Logβ under gauge action. The converse implication holds if (A, ρ, Σ) is mean ergodic.

(ii) Under the condition |β| = r_{ρ}, there exists a linear isomorphism between the space E_{β}(ρ) of eigenvectors of continuous linear functionals on A and the space KMS_{β}(O_{ρ}) of continuous linear functionals on O_{ρ} satisfying KMS condition at Logβ.

(iii) If (A, ρ, Σ) is uniquely ergodic, there uniquely exists a state on O_{ρ}

satisfying KMS condition at log r_{ρ}. (iv) If in particular (A, ρ, Σ) is mean ergodic, then dim E_{β}(ρ) ≤ 1 for all

β ∈ C.

In the proof of the above theorem, a Perron-Frobenius type theorem is proved (Theorem 3.13). Let D_{ρ} be the C^{∗}-subalgebra of O_{ρ} generated by all elements of the form: S_{α_{1}} ⋯ S_{α_{k}}xS_{α_{k}}^{∗} ⋯ S_{α_{1}}^{∗} for x ∈ A, α_{1}, …, α_{k} ∈ Σ. Let ϕ_{ρ} be the endomorphism on D_{ρ} defined by ϕ_{ρ}(y) = ∑_{α∈Σ} S_{α}yS_{α}^{∗}, y ∈ D_{ρ}, which comes from the left-shift on the underlying shift spase Λ_{ρ}. Suppose that (A, ρ, Σ) is uniquely ergodic. The restriction of the unique KMS state on O_{ρ} is not necessarily a ϕ_{ρ}-invariant state. We will clarify a relationship between KMS states on O_{ρ} and ϕ_{ρ}-invariant states on D_{ρ} as in the following way:

Theorem 1.2. Assume that (A, ρ, Σ) is irreducible and mean ergodic. Let τ be the restriction to D_{ρ} of the unique KMS state on O_{ρ} at logr_{ρ} and x_{ρ} be a positive element of A defined by the limit of the mean

lim_{n→∞} \frac{1}{n}(1 + T_{ρ}(1) + ⋯ + T_{ρ}^{n−1}(1)).

Let μ_{ρ} be a linear functional on D_{ρ} defined by

μ_{ρ}(y) = τ(yx_{ρ}), y ∈ D_{ρ}.

(i) μ_{ρ} is a faithful, ϕ_{ρ}-invariant and ergodic state on D_{ρ} in the sense that the formula

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} μ_{ρ}(ϕ_{ρ}^{k}(y)x) = μ_{ρ}(y)μ_{ρ}(x), x, y ∈ D_{ρ}

holds. (ii) μ_{ρ} gives rise to a unique ϕ_{ρ}-invariant probability measure absolutely continuous with respect to the probability measure for the state τ.

(iii) μ_{ρ} is equivalent to the state τ as a measure on D_{ρ}.

For a C^{∗}-symbolic dynamical system (A_{A}, ρ^{A}, Σ) coming from an irreducible matrix A = [A(i, j)]_{i,j=1}^{N} with entries in {0, 1}, the subalgebra D_{ρ^{A}} is nothing but the commutative C^{∗}-algebra C(X_{A}) of all continuous functions on the right one-sided topological Markov shift X_{A}. As ϕ_{ρ^{A}} corresponds to the leftshift σ_{A} on X_{A}, the above unique ϕ_{ρ^{A}} -invariant state τ is the Parry measure on X_{A}. The positive element x_{ρ^{A}} is given by the positive Perron eigenvector x_{ρ^{A}} = [x_{j}]_{j=1}^{N} of the transpose A^{t} of A satisfying ∑_{j=1}^{N} τ(S_{j}S_{j}^{∗})x_{j} = 1, where [τ(S_{j}S_{j}^{∗})]_{j=1}^{N} is the normalized Perron eigenvector of A. This paper is organized as follows: In Section 2, we will briefly review C^{∗}- symbolic dynamical systems and its C^{∗}-algebras O_{ρ}. In Section 3, we will study ergodic properties of the operator T_{ρ} : A ⟶ A and the eigenspace E_{β}(ρ) . In Section 4, we will study extendability of a linear functional belonging to E_{β}(ρ) to the subalgebra D_{ρ} of O_{ρ}, which will extend to O_{ρ}. In Section 5, we will prove Theorem 1.1. In Section 6, we will study a relationship between KMS states and ϕ_{ρ}-invariant states on D_{ρ} to prove Theorem 1.2. In Section 7, we will present several examples of continuous linear functionals on O_{ρ} satisfying KMS conditions.

2. C^{∗}-symbolic dynamical systems and their crossed products

Let A be a unital C^{∗}-algebra. In what follows, an endomorphism of A means a ∗-endomorphism of A that does not necessarily preserve the unit 1 of A. Denote by Z_{A} the center {x ∈ A | ax = xa for all a ∈ A} of A. Let Σ be a finite set. A finite family of nonzero endomorphisms ρ_{α}, α ∈ Σ of A indexed by elements of Σ is said to be essential if ρ_{α}(Z_{A}) ⊂ Z_{A} for α ∈ Σ and ∑_{α∈Σ} ρ_{α}(1) ≥ 1. If in particular, A is commutative, the family ρ_{α}, α ∈ Σ is essential if and only if ∑_{α∈Σ} ρ_{α}(1) ≥ 1. We remark that the definition in [31] of “essential” for ρ_{α}, α ∈ Σ is weaker than the above dfinition. It is said to be faithful if for any nonzero x ∈ A there exists a symbol α ∈ Σ such that ρ_{α}(x) ≠ 0. Definition ([31]). A C^{∗}-symbolic dynamical system is a triplet (A, ρ, Σ) consisting of a unital C^{∗}-algebra A and an essential, faithful finite family {ρ_{α}}_{α∈Σ} of endomorphisms of A. Two C^{∗}-symbolic dynamical systems (A, ρ, Σ) and (A′, ρ′, Σ′) are said to be isomorphic if there exist an isomorphism Φ : A → A′ and a bijection π : Σ → Σ′ such that Φ ∘ ρ_{α} = ρ′_{π(α)} ∘ Φ for all α ∈ Σ. For an automorphism α of a unital C^{∗}-algebra A, by setting Σ = {α}, ρ_{α} = α the triplet (A, ρ, Σ) becomes a C^{∗}- symbolic dynamical system. A C^{∗}-symbolic dynamical system (A, ρ, Σ) yields a subshift Λ_{ρ} over Σ such that a word α_{1} ⋯ α_{k} of Σ is admissible for Λ_{ρ} if and only if ρ_{α_{k}} ∘ ⋯ ∘ ρ_{α_{1}} ≠ 0 ([31, Proposition 2.1]). We say that a subshift Λ acts on a C^{∗}-algebra A if there exists a C^{∗}-symbolic dynamical system (A, ρ, Σ) such that the associated subshift Λ_{ρ} is Λ. For a C^{∗}-symbolic dynamical system (A, ρ, Σ) the C^{∗}-algebra O_{ρ} has been originally constructed in [31] as a C^{∗}-algebra from a Hilbert C^{∗}-bimodule by using a Pimsner’s general construction of Hilbert C^{∗}-bimodule algebras [40] (cf. [16] etc.). It is called the C^{∗}-symbolic crossed product of A by the subshift Λ_{ρ}, and realized as the universal C^{∗}-algebra C^{∗}(x, S_{α}; x ∈ A, α ∈ Σ) generated by x ∈ A and partial isometries S_{α}, α ∈ Σ subject to the following relations called (ρ):

∑_{γ∈Σ} S_{γ}S_{γ}^{∗} = 1,  S_{α}S_{α}^{∗}x = xS_{α}S_{α}^{∗},  S_{α}^{∗}xS_{α} = ρ_{α}(x)

for all x ∈ A and α ∈ Σ.

Let G = (G, λ) be a left-resolving finite labeled graph with underlying finite directed graph G = (V, E) and labeling map λ : E → Σ (see [28, p.76]). Denote by v_{1}, …, v_{N} the vertex set V. Assume that every vertex has both an incoming edge and an outgoing edge. Consider the N-dimensional commutative C^{∗}- algebra A_{G} = CE_{1} ⊕⋯⊕CE_{N} where each minimal projection E_{i} corresponds to the vertex v_{i} for i = 1, …, N. Define an N × N-matrix for α ∈ Σ by

A^{G}(i, α, j) = \begin{cases}1 & if there exists an edge e from v_{i} to v_{j} with λ(e) = α, \\ 0 & otherwise \end{cases} (2.1)

for i, j = 1, …, N. We set ρ_{α}^{G}(E_{i}) = ∑_{j=1}^{N} A^{G}(i, α, j)E_{j} for i = 1, …, N. Then ρ_{α}^{G}, α ∈ Σ define endomorphisms of A_{G} such that (A_{G}, ρ^{G}, Σ) is a C^{∗}- symbolic dynamical system for which the subshift Λ_{ρ^{G}} is the sofic shift Λ_{G} presented by G. Conversely, for a C^{∗}-symbolic dynamical system (A, ρ, Σ), if A is C^{N}, there exists a left-resolving labeled graph G such that A = A_{G} and Λ_{ρ} = Λ_{G} the sofic shift presented by G ([31, Proposition 2.2]). Put A_{G}(i, j) = ∑_{α∈Σ} A^{G}(i, α, j), i, j = 1, …, N. The N × N matrix A_{G} = [A_{G}(i, j)]_{i,j=1,...,N} is called the underlying nonnegative matrix for G. Consider the matrix A_{G}^{[2]} = [A_{G}^{[2]}(e, f)]_{e,f∈E} indexed by edges E whose entries are in {0, 1} by setting

A_{G}^{[2]}(e, f) = \begin{cases}1 & if f follows e, \\ 0 & otherwise. \end{cases}  (2.2)

The C^{∗}-algebra O_{ρ^{G}} for the C^{∗}-symbolic dynamical system (A_{G}, ρ^{G}, Σ) is the Cuntz-Krieger algebra O_{A_{G}^{[2]}} (cf. [30, Proposition 7.1], [1]). More generally let 𝔏 be a λ-graph system (V, E, λ, ι) over Σ. We equip each vertex set V_{l} with discrete topology. We denote by Ω_{𝔏} the compact Hausdorff space with dim Ω_{𝔏} = 0 of the projective limit V_{0} ←^{ι} V_{1} ←^{ι} V_{2} ←^{ι} ⋯ as in [30, Section 2]. Since the algebra C(V_{l}) denoted by A_{𝔏,l} of all continuous functions on V_{l} is the commutative finite dimensional algebra, the commutative C^{∗}-algebra C(Ω_{𝔏}) is an AF-algebra, that is denoted by A_{𝔏}. We then have a C^{∗}-symbolic dynamical system (A_{𝔏}, ρ^{𝔏}, Σ) such that the subshift Λ_{ρ^{𝔏}} coincides with the subshift Λ_{𝔏} presented by 𝔏. Conversely, for a C^{∗}-symbolic dynamical system (A, ρ, Σ), if the algebra A is C(X) with dim X = 0, there exists a λgraph system 𝔏 over Σ such that the associated C^{∗}-symbolic dynamical system (A_{𝔏}, ρ^{𝔏}, Σ) is isomorphic to (A, ρ, Σ) ([31, Theorem 2.4]). The C^{∗}-algebra O_{ρ^{𝔏}} is the C^{∗}-algebra O_{𝔏} associated with the λ-graph system 𝔏. Let α be an automorphism of a unital C^{∗}-algebra A. Put Σ = {α} and ρ_{α} = α. The C^{∗}-algebra O_{ρ} for the C^{∗}-symbolic dynamical system (A, ρ, Σ) is the ordinary C^{∗}-crossed product A ×_{α} Z.

In what follows, for a subset F of a C^{∗}-algebra B, we will denote by C^{∗}(F) the C^{∗}-subalgebra of B generated by F. Let (A, ρ, Σ) be a C^{∗}-symbolic dynamical system over Σ and Λ the associated subshift Λ_{ρ}. We denote by B_{k}(Λ) the set of admissible words μ of Λ with length |μ| = k. Put B_{∗}(Λ) = ∪_{k=0}^{∞}B_{k}(Λ), where B_{0}(Λ) consists of the empty word. Let S_{α}, α ∈ Σ be the partial isometries in O_{ρ} satisfying the relation (ρ). For μ = (μ_{1}, …, μ_{k}) ∈ B_{k}(Λ), we put S_{μ} = S_{μ_{1}} ⋯ S_{μ_{k}} and ρ_{μ} = ρ_{μ_{k}} ∘ ⋯ ∘ ρ_{μ_{1}}. In the algebra O_{ρ}, we set for k ∈ Z_{+},

D_{ρ}^{k} = C^{∗}(S_{μ}xS_{μ}^{∗} : μ ∈ B_{k}(Λ), x ∈ A),
D_{ρ} = C^{∗}(S_{μ}xS_{μ}^{∗} : μ ∈ B_{∗}(Λ), x ∈ A),
F_{ρ}^{k} = C^{∗}(S_{μ}xS_{ν}^{∗} : μ, ν ∈ B_{k}(Λ), x ∈ A) and
F_{ρ} = C^{∗}(S_{μ}xS_{ν}^{∗} : μ, ν ∈ B_{∗}(Λ), |μ| = |ν|, x ∈ A).

The identity S_{μ}xS_{ν}^{∗} = ∑_{α∈Σ} S_{μα}ρ_{α}(x)S_{να}^{∗} for x ∈ A, μ, ν ∈ B_{k}(Λ) holds so that the algebra F_{ρ}^{k} is embedded into the algebra F_{ρ}^{k+1} such that ∪_{k∈Z_{+}}F_{ρ}^{k} is dense in F_{ρ}. Similarly D_{ρ}^{k} is embedded into the algebra D_{ρ}^{k+1} such that ∪_{k∈Z_{+}}D_{ρ}^{k} is dense in D_{ρ}. The gauge action ρ̂ of the one-dimensional torus group T = {z ∈ C | |z| = 1} on O_{ρ} is defined by ρ̂_{z}(x) = x for x ∈ A and ρ̂_{z}(S_{α}) = zS_{α} for α ∈ Σ. The fixed point algebra of O_{ρ} under ρ̂ is denoted by (O_{ρ})ρ̂. Let E_{ρ} : O_{ρ} ⟶ (O_{ρ})ρ̂ be the conditional expectaton defined by

E_{ρ}(X) = ∫_{z∈T} ρ̂_{z}(X)dz,  X ∈ O_{ρ}.  (2.3)

It is routine to check that (O_{ρ})ρ̂ = F_{ρ}. Definition ([33]). A C^{∗}-symbolic dynamical system (A, ρ, Σ) satisfies condition (I) if there exists a unital increasing sequence

A_{0} ⊂ A_{1} ⊂ ⋯ ⊂ A

of C^{∗}-subalgebras of A such that ρ_{α}(A_{l}) ⊂ A_{l+1} for all l ∈ Z_{+}, α ∈ Σ, the union ∪_{l∈Z_{+}}A_{l} is dense in A and for ϵ > 0, k, l ∈ N with k ≤ l and X_{0} ∈ F_{ρ,l}^{k} = C^{∗}(S_{μ}xS_{ν}^{∗} : μ, ν ∈ B_{k}(Λ), x ∈ A_{l}), there exists an element g ∈ D_{ρ} ∩ A_{l}′(= {y ∈ D_{ρ} | ya = ay for a ∈ A_{l}}) with 0 ≤ g ≤ 1 such that

(i) ∥X_{0}ϕ_{ρ}^{k}(g)∥ ≥ ∥X_{0}∥ − ϵ, (ii) gϕ_{ρ}^{m}(g) = 0 for all m = 1, 2, …, k, where ϕ_{ρ}^{m}(X) = ∑_{μ∈B_{m}(Λ)} S_{μ}XS_{μ}^{∗}.

As the element g belongs to the diagonal subalgebra D_{ρ} of F_{ρ}, the condition (I) is intrinsically determined by (A, ρ, Σ) by virtue of [31, Lemma 4.1]. The condition (I) for (A, ρ, Σ) yields the uniqueness of the C^{∗}-algebra O_{ρ} under the relations (ρ) ([33]). If a λ-graph system 𝔏 over Σ satisfies condition (I), then (A_{𝔏}, ρ^{𝔏}, Σ) satisfies condition (I) (cf. [30, Lemma 4.1]). Recall that the positive operator λ_{ρ} : A ⟶ A is defined by λ_{ρ}(x) = ∑_{α∈Σ} ρ_{α}(x), x ∈ Σ. Then a C^{∗}-symbolic dynamical system (A, ρ, Σ) is said to be irreducible, if there exists no nontrivial ideal of A invariant under λ_{ρ}. It has been shown in [31] that if (A, ρ, Σ) satisfies condition (I) and is irreducible, then the C^{∗}-algebra O_{ρ} is simple. Interesting examples of (A, ρ, Σ) in [31], [34] which we have seen from the view point of symbolic dynamics come from ones for which A is commutative. Hence we assume that the algebra A is commutative so that A is written as C(Ω) for some compact Hausdorff space Ω henceforth. For the cases that A is noncommutative, our discussions in this paper well work by considering tracial states on A in stead of states on A under slight modifications.

3. Ergodicity and Perron-Frobenius type theorem

In this section, we will study ergodic properties of a C^{∗}-symbolic dynamical system (A, ρ, Σ) and prove a Perron-Frobenius type theorem. Let A^{∗} denote the Banach space of all complex valued continuous linear functionals on A. For β ∈ C with β ≠ 0, set

E_{β}(ρ) = {φ ∈ A^{∗} | φ ∘ λ_{ρ}(a) = βφ(a) for all a ∈ A}.

It is possible that E_{β}(ρ) is {0}. A nonzero continuous linear functional φ in E_{β}(ρ) is called an eigenvector of the operator λ_{ρ}^{∗} with respect to the eigenvalue β. Let r_{ρ} be the spectral radius of the positive operator λ_{ρ} : A ⟶ A. Since λ_{ρ}^{k}(1) ≥ 1, k ∈ N, one sees that r_{ρ} ≥ 1. As Sp(λ_{ρ}) = Sp(λ_{ρ}^{∗}) (cf. [8, VI. 2.7]), we note r_{ρ} = r(λ_{ρ}^{∗}). Let S(A) denote the state space of A.

Lemma 3.1. (A, ρ, Σ) is irreducible if and only if for a state φ on A and a nonzero element x ∈ A, there exists a natural number n such that φ(λ_{ρ}^{n}(x^{∗}x)) > 0.

Proof. Suppose that (A, ρ, Σ) is irreducible. For a state φ on A, put

I_{φ} = {x ∈ A|φ(λ_{ρ}^{n}(x^{∗}x)) = 0 for all n ∈ N}

which is an ideal of A because A is commutative. The Schwarz type inequality

λ_{ρ}^{n}(λ_{ρ}(x)^{∗}λ_{ρ}(x)) ≤ ∥λ_{ρ}∥λ_{ρ}^{n+1}(x^{∗}x) for x ∈ A

implies that I_{φ} is λ_{ρ}-invariant. Hence I_{φ} is trivial. Conversely, let I be an ideal of A invariant under λ_{ρ}. Put B = A/I. Denote by q : A ⟶ B the quotient map. Take ψ ∈ S(B) a state. Put φ = ψ ∘ q. For y ∈ I, as φ(λ_{ρ}^{n}(y^{∗}y)) = 0, n ∈ N, one sees that y = 0 and hence I = {0} by the hypothesis. Hence (A, ρ, Σ) is irreducible. □

We denote by T_{ρ} : A ⟶ A the positive operator \frac{1}{r_{ρ}} λ_{ρ}. The spectral radius of T_{ρ} is 1. A state τ on A is called an invariant state if τ ∘ T_{ρ} = τ on A, equivalently τ ∈ E_{r_{ρ}}(ρ).

Corollary 3.2. Suppose that (A, ρ, Σ) is irreducible. Then any positive eigenvector of λ_{ρ}^{∗} for a nonzero eigenvalue is faithful.

Proof. Let φ ∈ E_{β}(ρ) be a positive linear functional for some nonzero β ∈ C. Since φ(λ_{ρ}(1)) = βφ(1), one has β > 0. By the preceding lemma, one has φ(x^{∗}x) > 0 for nonzero x ∈ A. □

Yasuo Watatani has kindly informed to the author that the lemma below, which is seen from [41, Theorem 2.5], is needed in the proof of Lemma 3.4. In our restrictive situation, we may directly prove it as in the following way.

Lemma 3.3. The spectral radius r_{ρ} of the operator λ_{ρ} is contained in the spectrum Sp(λ_{ρ}) of λ_{ρ}.

Proof. The resolvent R(z) = (z − λ_{ρ})^{−1} for λ_{ρ} has the expansion R(z) = ∑_{n=0}^{∞} \frac{λ_{ρ}^{n}}{z^{n+1}} for z ∈ C, |z| > r_{ρ} which converges in norm. We note that the family {R(z)}_{|z|>r_{ρ}} is not uniformly bounded. Otherwise, there exists a constant M > 0 such that ∥R(z)∥ < M for z ∈ C, |z| > r_{ρ}. By the compactness of Sp(λ_{ρ}), we may find z_{∘} ∈ Sp(λ_{ρ}) with |z_{∘}| = r_{ρ}. Take z_{n} ∉ Sp(λ_{ρ}) satisfying lim_{n→∞} z_{n} = z_{∘} and |z_{n}| > r_{ρ}. The resolvent equation R(z_{n}) − R(z_{m}) = (z_{n} − z_{m})R(z_{n})R(z_{m}) implies the inequality ∥R(z_{n}) − R(z_{m})∥ ≤ |z_{n} − z_{m}|M^{2} so that there exists a bounded linear operator R_{∘} = lim_{n→∞} R(z_{n}) on A. The equality (z_{n}−λ_{ρ})R(z_{n})x = x, x ∈ A implies (z_{∘}−λ_{ρ})R_{∘}x = x, x ∈ A and hence z_{∘} ∉ Sp(λ_{ρ}) a contradiction. Thus there exists r_{n} ∈ C such that |r_{n}| ∉ Sp(λ_{ρ}) and |r_{n}| ↓ r_{ρ} and lim_{n→∞} ∥R(r_{n})f∥ = ∞ for some f ∈ A. We may assume that f ≥ 0. For a state φ on A, one has

|φ(R(r_{n})f)| ≤ ∑_{k=0}^{∞} \frac{φ(λ_{ρ}^{k}(f))}{|r_{n}|^{k+1}} = φ(R(|r_{n}|)f).

Denote by w(y) the numerical radius of an element y ∈ A, which is defined by

w(y) = sup{φ(y) | φ ∈ S(A)}.

As the inequalities \frac{1}{2}∥y∥ ≤ w(y) ≤ ∥y∥ always hold (cf. [13, p.95]), one sees

\frac{1}{2}∥R(r_{n})f∥ ≤ w(R(r_{n})f) ≤ w(R(|r_{n}|)f) ≤ ∥R(|r_{n}|)f∥

so that

lim_{n→∞} ∥R(|r_{n}|)f∥ = ∞.

If r_{ρ} ∉ Sp(λ_{ρ}), the condition |r_{n}| ∉ Sp(λ_{ρ}) means that R(|r_{n}|) ↑ R(r_{ρ}) because R(z) increases for z ↓ r_{ρ}. Hence R(|r_{n}|)f ↑ R(r_{ρ})f and lim_{n→∞} ∥R(|r_{n}|)f∥ = ∥R(r_{ρ})f∥ < ∞, a contradiction. Therefore we conclude r_{ρ} ∈ Sp(λ_{ρ}). □

The following lemma is crucial.

Lemma 3.4. Suppose that (A, ρ, Σ) is irreducible. Then there exists a faithful invariant state on A.

Proof. We denote by R^{∗}(t) the resolvent of λ_{ρ}^{∗} : A^{∗} → A^{∗} defined by R^{∗}(t)φ = (t − λ_{ρ}^{∗})^{−1}φ for φ ∈ A^{∗}, t > r(λ_{ρ}^{∗}). As r_{ρ} = r(λ_{ρ}^{∗}), there exists φ_{0} ∈ A^{∗} such that ∥R^{∗}(t)φ_{0}∥ is unbounded for t ↓ r_{ρ} by Lemma 3.3. We may assume that φ_{0} is a state on A. Put

φ_{n} = \frac{R^{∗}(r_{ρ} + \frac{1}{n})φ_{0}}{∥R^{∗}(r_{ρ} + \frac{1}{n})φ_{0}∥} for n = 1, 2, ....

Since R^{∗}(t) is positive for t > r_{ρ}, each φ_{n} is a state on A so that there exists a weak^{∗} cluster point φ_{∞} ∈ S(A) of the sequence {φ_{n}} in S(A). As we see

(r_{ρ} − λ_{ρ}^{∗})φ_{n} = −\frac{1}{n}φ_{n} + \frac{φ_{0}}{∥R^{∗}(r_{ρ} + \frac{1}{n})φ_{0}∥},

Documenta Mathematica 16 (2011) 133–175

we get r_{ρ}φ_{∞} = λ_{ρ}^{∗}φ_{∞} so that φ_{∞} ∈ E_{r_{ρ}}(ρ). By Corollary 3.2, one knows that φ_{∞} is faithful on A. □

Definition. A C^{∗}-symbolic dynamical system (A, ρ, Σ) is said to be uniquely ergodic if there exists a unique invariant state on A. Denote by τ the unique invariant state. If (A, ρ, Σ) is irreducible and uniquely ergodic, the unique invariant state τ is automatically faithful because any invariant state is faithful. There is an example of a C^{∗}-symbolic dynamical system (A, ρ, Σ) for which a unique invariant state is not faithful, unless (A, ρ, Σ) is irreducible. Let

A = C ⊕ C, Σ = \{1, 2\} and ρ_{1} = [\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}] , ρ_{2} = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] . Then λ_{ρ} = [\begin{matrix}2 & 0 \\ 0 & 1 \end{matrix}] ,

r_{ρ} = 2 and T_{ρ} = [\begin{matrix}1 & 0 \\ 0 & \frac{1}{2} \end{matrix}] . The vector [\begin{matrix}1 \\ 0 \end{matrix}] is a unique invariant state on A, that

is not faithful. We will see, in Section 7, that the C^{∗}-symbolic dynamical system (A_{G}, ρ^{G}, Σ) for a finite labeled graph G is uniquely ergodic if and only if the underlying nonnegative matrix A_{G} is irreducible. We will next consider the eigenvector space of the operator λ_{ρ} on A. We are assuming that the algebra A is commutative so that A is written as C(Ω) for some compact Hausdorff space Ω.

Lemma 3.5. Assume that (A, ρ, Σ) is irreducible.

(i) If T_{ρ} has a nonzero fixed element in A, then T_{ρ} has a nonzero positive fixed element in A,

(ii) A nonzero positive fixed element by T_{ρ} in A must be strictly positive. (iii) If there exist two nonzero positive fixed elements by T_{ρ} in A, then one

is a scalar multiple of the other. (iv) The dimension of the space consisting of the fixed elements by T_{ρ} is at

most one.

Proof. (i) Let y ∈ A be a nonzero fixed element by T_{ρ}. Since y^{∗} is also fixed by T_{ρ}, we may assume that y = y^{∗}. Denote by y = y^{+} − y^{−} with y^{+}, y^{−} ≥ 0 the Jordan decomposition of y. We have y^{+} ≥ y and hence T_{ρ}(y^{+}) ≥ T_{ρ}(y) = y. As T_{ρ}(y^{+}) ≥ 0, one sees that T_{ρ}(y^{+}) ≥ y^{+}. Now (A, ρ, Σ) is irreducible so that there exists a faithful invariant state τ on A. Since τ(T_{ρ}(y^{+}) − y^{+}) = 0, one has T_{ρ}(y^{+}) = y^{+}. Similarly we have T_{ρ}(y^{−}) = y^{−}. As y ≠ 0, either y^{+} or y^{−} is not zero. (ii) Let y ∈ A be a nonzero fixed positive element by T_{ρ}. Suppose that there exists ω_{0} ∈ Ω such that y(ω_{0}) = 0. Let I_{y} be the closed ideal of A generated by y. For a nonzero positive element f ∈ A we have

T_{ρ}(fy) ≤ ∥f∥T_{ρ}(y) = ∥f∥y

so that T_{ρ}(fy) belongs to I_{y}. As the ideal I_{y} is approximated by linear combinations of the elements of the form fy, f ∈ A, f ≥ 0, the ideal I_{y} is invariant under T_{ρ}. Now (A, ρ, Σ) is irreducible so that I_{y} = A. As any element of I_{y} vanishes at ω_{0}, a contradiction.

(iii) Let x, y ∈ A be nonzero positive fixed elements by T_{ρ}. By the above discussions, they are strictly positive. Set c_{0} = min\{\frac{x(ω)}{y(ω)} | ω ∈ Ω\}. The function x − c_{0}y is positive element but not strictly positive, so that it must be zero. (iv) Let y ∈ A be a fixed element under T_{ρ}, which is written as the Jordan decomposition y = y_{1}−y_{2}+i(y_{3}−y_{4}) for some positive elements y_{i}, i = 1, 2, 3, 4 in A. By the above discussions, all the elements y_{i}, i = 1, 2, 3, 4 are fixed under T_{ρ} and they are strictly positive if it is nonzero. Hence (iii) implies the desired assertion. □

Definition. A C^{∗}-symbolic dynamical system (A, ρ, Σ) is said to satisfy (FP) if there exists a nonzero fixed element in A under T_{ρ}. If in particular, (A, ρ, Σ) is irreducible, a nonzero fixed element can be taken as a strictly positive element in A by the previous lemma.

Lemma 3.6. Assume that (A, ρ, Σ) is irreducible and satisfies (FP).

(i) If there exists a state in E_{β}(ρ) for some β ∈ C with β ≠ 0, then we have β = r_{ρ}.

(ii) If in particular, (A, ρ, Σ) is uniquely ergodic, the eigenspace E_{r_{ρ}}(ρ) is of one-dimensional.

Proof. (i) Suppose that there exists a state ψ in E_{β}(ρ) for some β ∈ C with β ≠ 0. Let x_{0} ∈ A be a nonzero fixed element by T_{ρ}. One may take it to be strictly positive by the preceding lemma. Since λ_{ρ}(x_{0}) = r_{ρ}x_{0}, one has

βψ(x_{0}) = ψ(λ_{ρ}(x_{0})) = r_{ρ}ψ(x_{0}).

By Corollary 3.2, one has ψ(x_{0}) > 0 so that β = r_{ρ}. (ii) Take an arbitrary φ ∈ E_{r_{ρ}}(ρ). Put φ^{∗}(x) = \overline{φ(x^{∗})}, x ∈ A and hence φ^{∗} ∈ E_{r_{ρ}}(ρ). Both of the continuous linear functionals φ_{Re} = \frac{1}{2}(φ + φ^{∗}) and φ_{Im} = \frac{1}{2i}(φ − φ^{∗}) belong to E_{r_{ρ}}(ρ) which come from real valued measures on Ω. Put ψ = φ_{Re}. Let ψ = ψ_{+} − ψ_{−} be the Jordan decomposition of ψ, where ψ_{+}, ψ_{−} are positive linear functionals on A. Since ψ_{+} ≥ ψ, one has T_{ρ}^{∗}ψ_{+} ≥ T_{ρ}^{∗}ψ = ψ. As T_{ρ}^{∗}ψ_{+} is positive, one has T_{ρ}^{∗}ψ_{+} ≥ ψ_{+}. Now (A, ρ, Σ) is irreducible and satisfies (FP) so that one finds a strictly positive element x_{0} ∈ A fixed by T_{ρ}. Then ψ̃ = T_{ρ}^{∗}ψ_{+} − ψ_{+} is a positive linear functional satisfying ψ̃(x_{0}) = 0. It follows that ψ̃ = 0 so that T_{ρ}^{∗}ψ_{+} = ψ_{+}. Similarly we have T_{ρ}^{∗}ψ_{−} = ψ_{−}. As both ψ_{+}, ψ_{−} are positive linear functionals on A, the unique ergodicity asserts that there exist 0 ≤ c_{+}, c_{−} ∈ R such that ψ_{+} = c_{+}τ, ψ_{−} = c_{−}τ. By putting c_{Re} = c_{+} − c_{−}, one has φ_{Re} = c_{Re}τ and similarly φ_{Im} = c_{Im}τ for some real number c_{Im}. Therefore we have

φ = (c_{Re} + ic_{Im})τ.

Hence any continuous linear functional fixed by T_{ρ} is a scalar multiple of τ, so that

dim E_{r_{ρ}}(ρ) = 1.
□

A C^{∗}-symbolic dynamical system (A, ρ, Σ) is said to be power-bounded if the sequence {∥T_{ρ}^{k}∥ | k ∈ N} is bounded. As T_{ρ}^{k} : A ⟶ A is completely positive, the equalities ∥T_{ρ}^{k}∥ = ∥T_{ρ}^{k}(1)∥ = ∥\frac{1}{r_{ρ}^{k}} ∑_{μ∈B_{k}(Λ)} ρ_{μ}(1)∥ hold. We remark that for an irreducible matrix A = [A(i, j)]_{i,j=1}^{N} with entries in {0, 1}, the associated C^{∗}-symbolic dynamical system (A_{A}, ρ^{A}, Σ) defined in the Cuntz-Krieger algebra O_{A} is power-bounded. One indeed sees that there is a constant d > 0 such that

∑_{i,j=1}^{N} A^{k}(i, j) ≤ d · r_{A}^{k} (cf. [28, Proposition 4.2.1]).

Hence

∥λ_{A}^{k}(1)∥ = max_{i} ∑_{j=1}^{N} A^{k}(i, j) ≤ d · r_{A}^{k}.

Lemma 3.7. Assume that (A, ρ, Σ) is irreducible. If (A, ρ, Σ) satisfies (FP), then (A, ρ, Σ) is power-bounded.

Proof. As (A, ρ, Σ) is irreducible and satisfies (FP), there exists a strictly positive fixed element x_{0} of A under T_{ρ}. Since Ω is compact, one finds positive constants c_{1}, c_{2} such that 0 < c_{1} < x_{0}(ω) < c_{2} for all ω ∈ Ω. It follows that

c_{1}T_{ρ}^{n}(1) = T_{ρ}^{n}(c_{1}1) ≤ T_{ρ}^{n}(x_{0}) = x_{0} ≤ c_{2}, n ∈ N.

Thus we have ∥T_{ρ}^{n}∥ = ∥T_{ρ}^{n}(1)∥ ≤ \frac{c_{2}}{c_{1}} for n ∈ N. □

We define the mean operator M_{n} : A ⟶ A for n ∈ N by setting

M_{n}(a) = \frac{a + T_{ρ}(a) + T_{ρ}^{2}(a) + ⋯ + T_{ρ}^{n−1}(a)}{n},  a ∈ A.  (3.1)

Definition. A C^{∗}-symbolic dynamical system (A, ρ, Σ) is said to be mean ergodic if for a ∈ A the limit lim_{n→∞} M_{n}(a) exists in A under norm-topology. For a mean ergodic (A, ρ, Σ), the limit lim_{n→∞} M_{n}(1) exists in A under normtopology, which we denote by x_{ρ} ∈ A

Lemma 3.8. Assume that (A, ρ, Σ) is irreducible. For a mean ergodic (A, ρ, Σ), we have for a ∈ A,

lim_{n→∞} M_{n}(a) = lim_{n→∞} M_{n}(T_{ρ}(a)) = lim_{n→∞} T_{ρ}(M_{n}(a)).  (3.2)

In particular x_{ρ} is a nonzero positive element which satisfies x_{ρ} = T_{ρ}(x_{ρ}) and τ(x_{ρ}) = 1 for an invariant state τ ∈ E_{r_{ρ}}(ρ).

Proof. For a ∈ A, the equality T_{ρ}(M_{n}(a)) = M_{n}(T_{ρ}(a)) is clear. As

(n + 1)M_{n+1}(a) − nM_{n}(a) = T_{ρ}^{n}(a),

one has

\frac{1}{n}T_{ρ}^{n}(a) = M_{n+1}(a) − M_{n}(a) + \frac{1}{n}M_{n+1}(a)

so that lim_{n→∞} \frac{1}{n}T_{ρ}^{n}(a) = 0. By the equality

T_{ρ}(M_{n}(a)) − M_{n}(a) = \frac{1}{n}(T_{ρ}^{n}(a) − a)

we have

lim_{n→∞}(T_{ρ}(M_{n}(a)) − M_{n}(a)) = lim_{n→∞} \frac{1}{n}(T_{ρ}^{n}(a) − a) = 0.

Take a faithful invariant state τ on A, we have

τ(x_{ρ}) = lim_{n→∞} τ(M_{n}(1)) = τ(1) = 1.

□

Proposition 3.9. Assume that (A, ρ, Σ) is irreducible. If (A, ρ, Σ) is mean ergodic, there exists a faithful invariant state τ on A such taht

lim_{n→∞} M_{n}(a) = τ(a)x_{ρ},  a ∈ A.  (3.3)

Proof. For a ∈ A, the limit Φ(a) = lim_{n→∞} M_{n}(a) is fixed by T_{ρ} so that it is a scalar multiple of x_{ρ} by Lemma 3.5 (iv). One may put

Φ(a) = τ(a)x_{ρ} for some τ(a) ∈ C.

It is easy to see that τ : A ⟶ C is a state. As Φ(T_{ρ}(a)) = Φ(a), one sees τ(T_{ρ}(a)) = τ(a) for a ∈ A. Hence τ is an invariant state on A. Now (A, ρ, Σ) is irreducible, the invariant state is faithful. □

Hence the following corollary is clear.

Corollary 3.10. Assume that (A, ρ, Σ) is irreducible. Then the following two assertions are equivalent:

(i) (A, ρ, Σ) is mean ergodic.

(ii) There exist an invariant state τ on A and a positive element x_{0} ∈ A with τ(x_{0}) = 1 such that lim_{n→∞} M_{n}(a) = τ(a)x_{0} for a ∈ A.

In this case x_{0} is given by lim_{n→∞} M_{n}(1)(= x_{ρ}), and τ is faithful.

Theorem 3.11. Assume that (A, ρ, Σ) is irreducible. Then the following two assertions are equivalent:

(i) (A, ρ, Σ) is mean ergodic.

(ii) (A, ρ, Σ) is uniquely ergodic and satisfies (FP).

Proof. (i) ⇒ (ii): Suppose that (A, ρ, Σ) is mean ergodic. Put Φ(a) = lim_{n→∞} M_{n}(a) for a ∈ A. The element x_{ρ} = Φ(1) is a nonzero fixed element of A under T_{ρ}. By the previous corollary, there exists an invariant state τ on A satisfying Φ(a) = τ(a)x_{ρ} for a ∈ A. For any invariant state ψ on A, we have ψ ∘ M_{n}(a) = ψ(a) for a ∈ A. Hence ψ(Φ(a)) = ψ(a) so that ψ(a) = ψ(τ(a)x_{ρ}) = τ(a)ψ(x_{ρ}). Since ψ(x_{ρ}) = 1, we obtain ψ(a) = τ(a). Therefore ψ = τ so that (A, ρ, Σ) is uniquely ergodic. (ii) ⇒ (i): Suppose that (A, ρ, Σ) is uniquely ergodic and satisfies (FP). By Lemma 3.7, (A, ρ, Σ) is power-bounded. Hence the sequence \{\frac{1}{n} ∑_{k=0}^{n−1} T_{ρ}^{k}\}_{n∈N} is uniformly bounded. This means that T_{ρ} : A ⟶ A is Cesàro bounded (cf.

[22, p.72]). As lim_{n→∞} \frac{T_{ρ}^{n−1}(a)}{n} = 0 for a ∈ A, the operator T_{ρ} : A → A satisfies the assumption of [22, p.74 Theorem 1.4]. To prove mean ergodicity, it suffices to show that F = {x ∈ A | T_{ρ}x = x} separates F^{∗} = {φ ∈ A^{∗} | φ ∘ T_{ρ} = φ}. By Lemma 3.6, one knows that F^{∗} = Cτ, where τ is a unique faithful invariant state on A. Hence if φ = cτ ∈ F^{∗} is nonzero, then c ≠ 0 and φ(x_{ρ}) = cτ(x_{ρ}) = c ≠ 0. This implies that F separates F^{∗}. Thus by [22, p.74 Theorem 1.4], (A, ρ, Σ) is mean ergodic. □

Remark. In [22, p.179], it is shown that a mean ergodic irreducible “Markov operator ”is uniquely ergodic. In our situation, the operator T_{ρ} does not necessarily satisfy T_{ρ}(1) = 1. Hence the operator T_{ρ} is not necessarily a Markov operator.

We summarize results obtained in this section as in the following way:

Theorem 3.12. Assume that (A, ρ, Σ) is irreducible. Then the following implications hold:

(ME) ⟺ (UE) + (FP) ⟹ (FP) ⟹ (PB)
⇓
dim E_{r_{ρ}}(ρ) = 1 ⟹ (UE),

where (ME) means mean ergodic, (UE) means uniquely ergodic, and (PB) means power-bounded.

If in particular (A, ρ, Σ) is irreducible and mean ergodic, the following Perron-Frobenius type theorem holds.

Theorem 3.13. Assume that (A, ρ, Σ) is irreducible and mean ergodic.

(i) There exists a unique pair of a faithful state τ on A and a strictly positive element x_{ρ} in A satisfying the conditions:

τ ∘ λ_{ρ} = r_{ρ}τ, λ_{ρ}(x_{ρ}) = r_{ρ}x_{ρ} and τ(x_{ρ}) = 1,

where r_{ρ} is the spectral radius of the positive operator λ_{ρ} on A. (ii) If there exists a continuous linear functional ψ on A satisfying

ψ ∘ λ_{ρ} = r_{ρ}ψ,

then ψ = cτ for some complex number c ∈ C. (iii) If there exists a state φ on A and a complex number β ∈ C with β ≠ 0

satisfying

φ ∘ λ_{ρ} = βφ,

then φ = τ and β = r_{ρ}.

(iv) For any a ∈ A, the limit lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} \frac{λ_{ρ}^{k}(a)}{r_{ρ}^{k}} exists in A in the norm topology such that

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} \frac{λ_{ρ}^{k}(a)}{r_{ρ}^{k}} = τ(a)x_{ρ}.

Proof. Under the assumption that (A, ρ, Σ) is irreducible, mean ergodicity is equivalent to unique ergodicity with (FP). (i) and (iv) follows from Corollary 3.10 and unique ergodicity. (ii) follows from Lemma 3.6 (ii). (iii) follows from Lemma 3.6 (i) and unique ergodicity. □

4. Extension of eigenvectors to F_{ρ}

In this section, we will study extendability of an eigenvector in E_{β}(ρ) to the subalgebra F_{ρ}. We fix a C^{∗}-symbolic dynamical system (A, ρ, Σ) satisfying condition (I) henceforth.

Lemma 4.1. Fix a nonnegative integer k ∈ Z_{+}. For any element x ∈ F_{ρ}^{k} there uniquely exists x_{μ,ν} in A for each μ, ν ∈ B_{k}(Λ) such that

x = ∑_{μ,ν∈B_{k}(Λ)} S_{μ}x_{μ,ν}S_{ν}^{∗}  and  x_{μ,ν} = ρ_{μ}(1)x_{μ,ν}ρ_{ν}(1). (4.1)

If in particular x belongs to D_{ρ}^{k}, there uniquely exists x_{μ} in A for each μ ∈ B_{k}(Λ) such that

x = ∑_{μ∈B_{k}(Λ)} S_{μ}x_{μ}S_{μ}^{∗}  and  x_{μ} = ρ_{μ}(1)x_{μ}ρ_{μ}(1).  (4.2)

Proof. For an element x in F_{ρ}^{k} and μ, ν ∈ B_{k}(Λ), put x_{μ,ν} = S_{μ}^{∗}xS_{ν} that belongs to A and satisfies the equalities (4.1). □

We set

D_{ρ}^{alg} = the algebraic linear span of S_{μ}aS_{μ}^{∗} for μ ∈ B_{∗}(Λ), a ∈ A, and
F_{ρ}^{alg} = the algebraic linear span of S_{μ}aS_{ν}^{∗} for μ, ν ∈ B_{∗}(Λ), |μ| = |ν|, a ∈ A.

Hence D_{ρ}^{alg} = ∪_{k=0}^{∞}D_{ρ}^{k} and F_{ρ}^{alg} = ∪_{k=0}^{∞}F_{ρ}^{k}. They are dense ∗-subalgebras of D_{ρ} and F_{ρ} respectively.

Lemma 4.2. For β ∈ C with |β| > 1 and φ ∈ E_{β}(ρ) on A, put

φ̃(S_{μ}aS_{μ}^{∗}) = \frac{1}{β^{|μ|}} φ(aρ_{μ}(1)),  a ∈ A, μ ∈ B_{∗}(Λ).  (4.3)

Then φ̃ is a well-defined (not necessarily continuous) linear functional on D_{ρ}^{alg}, that is an extension of φ.

Proof. By the expansion (4.2) for an element x ∈ D_{ρ}^{k}, the following definition of φ_{k}(x) yields a linear functional φ_{k} on D_{ρ}^{k}

φ_{k}(x) = ∑_{μ∈B_{k}(Λ)} \frac{1}{β^{k}} φ(x_{μ}).  (4.4)

We will show that φ_{k} = φ_{k+1} on D_{ρ}^{k}. As S_{μ}x_{μ}S_{μ}^{∗} = ∑_{α∈Σ} S_{μα}ρ_{α}(x_{μ})S_{μα}^{∗} and ρ_{μα}(1)ρ_{α}(x_{μ})ρ_{μα}(1) = S_{α}^{∗}ρ_{μ}(1)x_{μ}ρ_{μ}(1)S_{α} = ρ_{α}(x_{μ}), the following expression of x in D_{ρ}^{k+1}

x = ∑_{μ∈B_{k}(Λ),α∈Σ} S_{μα}ρ_{α}(x_{μ})S_{μα}^{∗}

is the unique expression of (4.2). Hence we obtain

φ_{k+1}(x) = ∑_{μ∈B_{k}(Λ),α∈Σ} \frac{1}{β^{k+1}} φ(ρ_{α}(x_{μ})) = \frac{1}{β^{k}} ∑_{μ∈B_{k}(Λ)} φ(x_{μ}) = φ_{k}(x).

The family {φ_{k}}_{k∈Z_{+}} of linear functionals on the subalgebras {D_{ρ}^{k}}_{k∈Z_{+}} yields a linear functional on the algebra D_{ρ}^{alg}. We denote it by φ̃. As the expansion a = ∑_{α∈Σ} S_{α}ρ_{α}(a)S_{α}^{∗} for a ∈ A is the unique expansion of a in (4.2) as an element of D_{ρ}^{1}, we have φ̃(a) = \frac{1}{β} ∑_{α∈Σ} φ(ρ_{α}(a)) = φ(a) so that φ̃ = φ on A. □

We will extend λ_{ρ} on A to F_{ρ} such as

λ_{ρ}(x) = ∑_{α∈Σ} S_{α}^{∗}xS_{α} for x ∈ F_{ρ}.

Lemma 4.3. Let ψ be a linear functional on F_{ρ}^{alg} such that its restriction to A is continuous. Then the following three conditions are equivalent:

(i) ψ is tracial and ψ ∘ λ_{ρ}(x) = βψ(x) for x ∈ F_{ρ}^{alg}.

(ii) ψ(S_{μ}xS_{ν}^{∗}) = δ_{μ,ν} \frac{1}{β^{|μ|}} ψ(xS_{μ}^{∗}S_{μ}) for x ∈ F_{ρ}^{alg}, μ, ν ∈ B_{∗}(Λ) with |μ| =

|ν|. (iii) There exists φ ∈ E_{β}(ρ) such that

ψ(S_{μ}aS_{ν}^{∗}) = δ_{μ,ν} \frac{1}{β^{|μ|}} φ(aρ_{μ}(1)) for a ∈ A, μ, ν ∈ B_{∗}(Λ)with |μ| = |ν|.

Proof. (i) ⇒ (ii): The equation (i) implies that for k ∈ N,

ψ(x) = \frac{1}{β^{k}} ∑_{γ∈B_{k}(Λ)} ψ(S_{γ}^{∗}xS_{γ}), x ∈ F_{ρ}^{alg}.

It then follows that for μ, ν ∈ B_{k}(Λ)

ψ(S_{μ}xS_{ν}^{∗}) = \frac{1}{β^{k}} ∑_{γ∈B_{k}(Λ)} ψ(S_{γ}^{∗}S_{μ}xS_{ν}^{∗}S_{γ}) = δ_{μ,ν} \frac{1}{β^{k}} ψ(xS_{μ}^{∗}S_{μ}).

(ii) ⇒ (iii): Define a linear functional φ on A by the restriction of ψ to the subalgebra A. By the equation (ii) for a ∈ A and hence S_{α}^{∗}aS_{α} ∈ A, we see

ψ(S_{α}S_{α}^{∗}a) = ψ(S_{α}S_{α}^{∗}aS_{α}S_{α}^{∗}) = \frac{1}{β} ψ(S_{α}^{∗}aS_{α}S_{α}^{∗}S_{α}) = \frac{1}{β} ψ(S_{α}^{∗}aS_{α})

so that φ ∈ E_{β}(ρ). The equation (iii) is clear. (iii) ⇒ (i): We will see that ψ is tracial. Let x, y ∈ F_{ρ}^{k} be expanded as in (4.1) so that x = ∑_{μ,ν∈B_{k}(Λ)} S_{μ}x_{μ,ν}S_{ν}^{∗}, y = ∑_{μ,ν∈B_{k}(Λ)} S_{μ}y_{μ,ν}S_{ν}^{∗}. We have

xy = ∑_{μ,ν,γ∈B_{k}(Λ)} S_{μ}x_{μ,ν}ρ_{ν}(1)y_{ν,γ}S_{γ}^{∗} = ∑_{μ,γ∈B_{k}(Λ)} S_{μ}( ∑_{ν∈B_{k}(Λ)} x_{μ,ν}y_{ν,γ})S_{γ}^{∗}

and ∑_{ν∈B_{k}(Λ)} x_{μ,ν}y_{ν,γ} = ρ_{μ}(1)(∑_{ν∈B_{k}(Λ)} x_{μ,ν}y_{ν,γ})ρ_{γ}(1), similarly

yx = ∑_{η,ν∈B_{k}(Λ)} S_{η}( ∑_{γ∈B_{k}(Λ)} y_{η,γ}x_{γ,ν})S_{ν}^{∗}

and ∑_{γ∈B_{k}(Λ)} y_{η,γ}x_{γ,ν} = ρ_{η}(1)(∑_{ν∈B_{k}(Λ)} y_{η,γ}x_{γ,ν})ρ_{ν}(1). It follows that

ψ(xy) = ∑_{μ,ν∈B_{k}(Λ)} \frac{1}{β^{k}} φ(x_{μ,ν}y_{ν,μ}) = ∑_{γ,η∈B_{k}(Λ)} \frac{1}{β^{k}} φ(y_{η,γ}x_{γ,η}) = ψ(yx).

Hence ψ is tracial on F_{ρ}^{k}. We will finally show that the equality in (i) holds. For S_{μ}aS_{ν}^{∗} ∈ F_{ρ}^{k} with a ∈ A, μ = (μ_{1}, …, μ_{k}), ν = (ν_{1}, …, ν_{k}) ∈ B_{k}(Λ), put μ_{[2,k]} = (μ_{2}, …, μ_{k}), ν_{[2,k]} = (ν_{2}, …, ν_{k}) ∈ B_{k−1}(Λ). One has

∑_{α∈Σ} ψ(S_{α}^{∗}(S_{μ}aS_{ν}^{∗})S_{α})

=δ_{μ_{1},ν_{1}}ψ(ρ_{μ_{1}}(1)S_{μ_{[2,k]}}aS_{ν_{[2,k]}}^{∗}ρ_{ν_{1}}(1))
=δ_{μ_{1},ν_{1}}ψ(S_{μ_{[2,k]}}S_{μ_{[2,k]}}^{∗}ρ_{μ_{1}}(1)S_{μ_{[2,k]}}aS_{ν_{[2,k]}}^{∗}ρ_{ν_{1}}(1)S_{ν_{[2,k]}}S_{ν_{[2,k]}}^{∗})
=δ_{μ_{1},ν_{1}}ψ(S_{μ_{[2,k]}}ρ_{μ}(1)aρ_{ν}(1)S_{ν_{[2,k]}}^{∗})
=δ_{μ_{1},ν_{1}}δ_{μ_{[2,k]},ν_{[2,k]}} \frac{1}{β^{k−1}} φ(ρ_{μ}(1)aρ_{ν}(1)ρ_{ν_{[2,k]}}(1))
=δ_{μ,ν} \frac{1}{β^{k−1}} φ(ρ_{μ}(1)aρ_{ν}(1))
=βψ(S_{μ}aS_{ν}^{∗}).
□

Let E_{D} : F_{ρ} ⟶ D_{ρ} denote the expectation satisfying

E_{D}(S_{μ}aS_{ν}^{∗}) = δ_{μ,ν}S_{μ}aS_{μ}^{∗}, a ∈ A, μ, ν ∈ B_{∗}(Λ), |μ| = |ν|.

Once we have an extension φ̃ to D_{ρ} of φ ∈ E_{β}(ρ), φ̃ has a further extension to F_{ρ} by φ̃ ∘ E_{D}. The extension φ̃ ∘ E_{D} on F_{ρ} is continuous if φ̃ is so on D_{ρ}. It satisfies

φ̃ ∘ E_{D}(S_{μ}aS_{ν}^{∗}) = δ_{μ,ν} \frac{1}{β^{|μ|}} φ(aρ_{μ}(1))  (4.5)

for a ∈ A, μ, ν ∈ B_{∗}(Λ) with |μ| = |ν|. Hence the extension of a continuous linear functional on D_{ρ} to F_{ρ} is automatic. We have only to study extension of a linear functional φ ∈ E_{β}(ρ) on A to D_{ρ}. The condition (iii) of Lemma 4.3 is equivalent to ψ = φ̃ ∘ E_{D} where φ̃ is a linear functional on D_{ρ}^{alg} obtained from φ ∈ E_{β}(ρ) as in Lemma 4.2, and so thst ψ is continuous if and only if φ̃ is continuous. We call the extension φ̃ on D_{ρ}^{alg} of φ ∈ E_{β}(ρ) the canonical extension of φ.

Lemma 4.4. Suppose that (A, ρ, Σ) is irreducible and power-bounded. For β ∈ C with |β| = r_{ρ} > 1, a (not necessarily positive) continuous linear functional φ ∈ E_{β}(ρ) on A extends to a continuous linear functional φ̃ on D_{ρ} satisfying (4.3).

Proof. As (A, ρ, Σ) is irreducible, we may take a faithful invariant state τ on A, which we will fix. By the hypothesis that (A, ρ, Σ) is power-bounded, there

exists a positive number M such that \frac{∥λ_{ρ}^{k}(1)∥}{r_{ρ}^{k}} < M for all k ∈ N. By [43, Theorem 4.2], there exists a partial isometry v ∈ A^{∗∗} and a positive linear functional ψ ∈ A^{∗} such that

φ(a) = ψ(av), a ∈ A.

For x = ∑_{μ∈B_{k}(Λ)} S_{μ}x_{μ}S_{μ}^{∗} ∈ D_{ρ}^{k} as in (4.2). Define a linear functional φ_{k} on D_{ρ}^{k} by (4.4). As in Lemma 4.2, φ_{k+1}|_{D_{ρ}^{k}} = φ_{k} and hence {φ_{k}}_{k∈N} defines a linear functional on D_{ρ}^{alg}. It then follows that

|φ(x_{μ})| = |ψ(ρ_{μ}(1)x_{μ}ρ_{μ}(1)v)| ≤ ψ(ρ_{μ}(1))^{\frac{1}{2}} ∥x_{μ}^{∗}x_{μ}∥^{\frac{1}{2}} ψ(v^{∗}ρ_{μ}(1)v)^{\frac{1}{2}} .

Since ρ_{μ}(1) commutes with v and

∥x_{μ}∥ = ∥S_{μ}x_{μ}S_{μ}^{∗}∥ ≤ max_{ν∈B_{k}(Λ)} ∥S_{ν}x_{ν}S_{ν}^{∗}∥ = ∥x∥,  (4.6)

we have

|φ(x_{μ})| ≤ ∥x∥ψ(ρ_{μ}(1))

and hence

|φ_{k}(x)| ≤ \frac{1}{|β|^{k}} ∑_{μ∈B_{k}(Λ)} |φ(x_{μ})| ≤ \frac{1}{|β|^{k}} ∥x∥ψ(λ_{ρ}^{k}(1)) = \frac{∥λ_{ρ}^{k}(1)∥}{r_{ρ}^{k}}ψ(1)∥x∥.

Therefore we have

|φ_{k}(x)| ≤ Mψ(1)∥x∥, x ∈ D_{ρ}^{k}

and hence {φ_{k}}_{k∈N} extends to a continuous linear functional on the closure D_{ρ} of D_{ρ}^{alg}. □

If in particular a linear functional φ ∈ E_{β}(ρ) is positive on A, it always extends to a continuous linear functionl on D_{ρ} as in the following way:

Lemma 4.5. Let β ∈ C be |β| > 1. If φ ∈ E_{β}(ρ) is a positive linear functional on A, then β becomes a positive real number and the canonical extension φ̃ to D_{ρ} is continuous on D_{ρ}.

Proof. One may assume that φ ≠ 0 and φ(1) = 1. We have β = βφ(1) = φ(λ_{ρ}(1)) ≥ 1. For k ∈ N, define a linear functional φ_{k} on D_{ρ}^{k} by (4.4). Since for x = ∑_{μ∈B_{k}(Λ)} S_{μ}x_{μ}S_{μ}^{∗} ∈ D_{ρ}^{k} we have by (4.6),

|φ(ρ_{μ}(1)x_{μ}ρ_{μ}(1))| ≤ φ(ρ_{μ}(1))^{\frac{1}{2}} φ(ρ_{μ}(1)x_{μ}^{∗}x_{μ}ρ_{μ}(1))^{\frac{1}{2}} ≤ ∥x∥φ(ρ_{μ}(1)),

it follows that

|φ_{k}(x)| ≤ \frac{1}{|β|^{k}} ∑_{μ∈B_{k}(Λ)} |φ(ρ_{μ}(1)x_{μ}ρ_{μ}(1))| ≤ \frac{1}{|β|^{k}} ∥x∥φ(λ_{ρ}^{k}(1)) = ∥x∥.

Therefore {φ_{k}}_{k∈N} extends to a state on D_{ρ}. □

We are now assuming that (A, ρ, Σ) is irreducible. By Lemma 3.4, there exists a faithful invariant state τ ∈ E_{r_{ρ}}(ρ) on A. By the previous lemma, the canonical extension τ̃ is continuous on D_{ρ} which satisfies

τ̃(S_{μ}aS_{μ}^{∗}) = \frac{1}{r_{ρ}^{|μ|}} τ(aρ_{μ}(1)),  a ∈ A, μ ∈ B_{∗}(Λ).  (4.7)

Lemma 4.6. For a faithful invariant state τ ∈ E_{r_{ρ}}(ρ) on A, the canonical extension τ̃ is faithful on D_{ρ}.

Proof. Suppose that τ̃ is not faithful on D_{ρ}. Put

Iτ̃ = {x ∈ D_{ρ} | τ̃(x^{∗}x) = 0}.

Since τ̃ is tracial on D_{ρ}, Iτ̃ is a nonzero ideal of D_{ρ}. By Lemma 4.3, the equality τ̃ ∘ λ_{ρ} = r_{ρ}τ̃ holds on D_{ρ} so that Iτ̃ is λ_{ρ}-invariant. The sequence D_{ρ}^{k}, k ∈ N of algebras is increasing such that ∪_{k∈N}D_{ρ}^{k} is dense in D_{ρ}. We may find k ∈ N such that Iτ̃ ∩ D_{ρ}^{k} ≠ 0. It is easy to see that λ_{ρ}^{k}(D_{ρ}^{k}) ⊂ A so that there exists a nonzero positive element x ∈ Iτ̃∩D_{ρ}^{k} such that λ_{ρ}^{k}(x) ∈ Iτ̃∩A. Hence Iτ̃∩A is a nonzero λ_{ρ}-invariant ideal of A. By the hypothesis that (A, ρ, Σ) is irreducible, we have a contradiction. □

For a faithful invariant state τ on A, we will write the canonical extension τ̃ of τ to D_{ρ} as still τ. Define a unital endomorphism ϕ_{ρ} : D_{ρ} ⟶ D_{ρ} by setting

ϕ_{ρ}(y) = ∑_{α∈Σ} S_{α}yS_{α}^{∗},  y ∈ D_{ρ}.  (4.8)

It induces a unital endomorphism on the enveloping von Neumann algebra D_{ρ}^{∗∗} of D_{ρ}, which we still denote by ϕ_{ρ}. The restriction of the positive map λ_{ρ} on F_{ρ} to D_{ρ} similarly induces a positive map on D_{ρ}^{∗∗}. We then need the following lemma for further discussions.

Lemma 4.7. The equality

λ_{ρ}(xϕ_{ρ}(y)) = λ_{ρ}(x)y,  x, y ∈ D_{ρ}^{∗∗}  (4.9)

holds.

Proof. Since D_{ρ} is dense in D_{ρ}^{∗∗} under σ(D_{ρ}^{∗∗}, D_{ρ}^{∗})-topology, it suffices to show the equality (4.9) for x, y ∈ D_{ρ}. One has

λ_{ρ}(xϕ_{ρ}(y)) = ∑_{α,γ∈Σ} S_{α}^{∗}xS_{γ}yS_{γ}^{∗}S_{α}

= ∑_{α∈Σ} S_{α}^{∗}xS_{α}yS_{α}^{∗}S_{α} = ∑_{α∈Σ} S_{α}^{∗}xS_{α}y = λ_{ρ}(x)y.

□

Recall that for a continuous linear functional ψ on a C^{∗}-algebra B there exist a partial isometry v ∈ B^{∗∗} and a positive linear functional |ψ| ∈ B^{∗} in a unique way such that

v^{∗}v = s(|ψ|),  ψ(x) = |ψ|(xv) for x ∈ B,  (4.10)

where s(|ψ|) denotes the support projection of |ψ| (cf. [43, Theorem 4.2]). The decomposition (4.10) is called the polar decomposition of ψ. The linear functional ψ : x ⟶ |ψ|(xv) is denoted by v|ψ|.

Lemma 4.8. Let β = re^{iθ} ∈ C be r, θ ∈ R with r > 1. For a (not necessarily positive) linear functional φ ∈ E_{β}(ρ) on A, let φ̃ be the extension on D_{ρ}^{alg} satisfying (4.3). Suppose that the linear functional φ̃ extends to a continuous linear functional on D_{ρ}. Denote by φ̃ = v|φ̃| its polar decomposition for a partial isometry v ∈ D_{ρ}^{∗∗} and a positive linear functional |φ̃| on D_{ρ} such that v^{∗}v = s(|φ̃|). Then we have

ϕ_{ρ}(v) = e^{iθ}v, |φ̃|(λ_{ρ}(x)) = r|φ̃|(x) for x ∈ D_{ρ}.

Hence the restriction of |φ̃| to A belongs to E_{r}(ρ) and |φ̃| satisfies

|φ̃|(S_{μ}aS_{μ}^{∗}) = \frac{1}{r^{|μ|}} |φ̃|(aρ_{μ}(1)), a ∈ A, μ ∈ B_{∗}(Λ).

Proof. Put a positive linear functional ψ on D_{ρ} and a partial isometry u in D_{ρ}^{∗∗} by setting

ψ(x) = \frac{1}{r}|φ̃|(λ_{ρ}(x)) for x ∈ D_{ρ} and u = e^{−iθ}ϕ_{ρ}(v).

As λ_{ρ}(xu) = e^{−iθ}λ_{ρ}(x)v for x ∈ D_{ρ} by Lemma 4.7. It follows that for x ∈ D_{ρ}

(uψ)(x) = \frac{1}{r}|φ̃|(λ_{ρ}(xu)) = \frac{1}{β} |φ̃|(λ_{ρ}(x)v) = φ̃(x).

Hence we have

φ̃ = uψ on D_{ρ}.

We will next show that s(ψ) = u^{∗}u. For y ∈ D_{ρ}, we have by Lemma 4.7

ψ(yu^{∗}u) = \frac{1}{r}|φ̃|(λ_{ρ}(yu^{∗}u)) = \frac{1}{r}|φ̃|(λ_{ρ}(yϕ_{ρ}(v^{∗}v))) = \frac{1}{r}|φ̃|(λ_{ρ}(y)v^{∗}v) = ψ(y).

Hence we have u^{∗}u ≥ s(ψ). On the other hand, suppose that a projection p ∈ D_{ρ}^{∗∗} satisfies

ψ(yp) = ψ(y) for y ∈ D_{ρ}.

We then have |φ̃|(λ_{ρ}(y(1 − p))) = 0 for all y ∈ D_{ρ}. For y = S_{α}S_{α}^{∗}, α ∈ Σ, one has |φ̃|(S_{α}^{∗}(1 − p)S_{α})) = 0. As S_{α}^{∗}(1 − p)S_{α} is a projection in D_{ρ}, one obtains that S_{α}^{∗}(1 − p)S_{α} ≤ 1 − v^{∗}v so that 1 − p ≤ 1 − ϕ_{ρ}(v^{∗}v). This implies that u^{∗}u ≤ p. Therefore we have u^{∗}u ≤ s(ψ) and hence

u^{∗}u = s(ψ).

By the uniqueness of the polar decomposition, we conclude that

v = u  and  |φ̃| = ψ on D_{ρ}

so that

ϕ_{ρ}(v) = e^{iθ}v, |φ̃|(λ_{ρ}(x)) = r|φ̃|(x) for x ∈ D_{ρ}.
□

Therefore we have

Theorem 4.9. Suppose that (A, ρ, Σ) is irreducible and power-bounded. For β ∈ C with |β| > 1, a (not necessarily positive) linear functional φ ∈ E_{β}(ρ) on A extends to D_{ρ} as a continuous linear functional φ̃ satisfying

φ̃(S_{μ}aS_{μ}^{∗}) = \frac{1}{β^{|μ|}} φ(aρ_{μ}(1)), a ∈ A, μ ∈ B_{∗}(Λ)

if |β| = r_{ρ}. If in particular, (A, ρ, Σ) is mean ergodic, the converse implication holds.

Proof. The first part of the assertions is direct from Lemma 4.4. Under the condition that (A, ρ, Σ) is mean ergodic, assume that the canonical extension φ̃ is continuous on D_{ρ}. The preceding lemma says that the positive linear functional |φ̃| belongs to E_{|β|}(ρ). Since the mean ergodicity implies (FP), by Lemma 3.6 (i) we see that |β| = r_{ρ}. □

Let us now assume that (A, ρ, Σ) is irreducible and satisfies dim E_{r_{ρ}}(ρ) = 1, and hence it is uniquely ergodic. Take a unique invariant state τ on A and denote still by τ its canonical extension on D_{ρ}. Denote by p_{τ} ∈ D_{ρ}^{∗∗} its support projection.

Lemma 4.10. Let w ∈ D_{ρ}^{∗∗} be a partial isometry satisfying

w^{∗}w = p_{τ}  and  ϕ_{ρ}(w) = w.  (4.11)

Then w is a scalar multiple of the projection p_{τ}.

Proof. Put wτ(x) = τ(xw) for x ∈ D_{ρ} and hence wτ ∈ D_{ρ}^{∗}. Since λ_{ρ}(x)w = λ_{ρ}(xϕ_{ρ}(w)) = λ_{ρ}(xw) by Lemma 4.7, it follows that for x ∈ D_{ρ}

wτ(λ_{ρ}(x)) = τ(λ_{ρ}(xw)) = r_{ρ}τ(xw) = r_{ρ}wτ(x).

In particular, we have wτ ∈ E_{r_{ρ}}(ρ). As dim E_{r_{ρ}}(ρ) = 1 by hypothesis, wτ is a scalar multiple of τ. Hence there exists c ∈ C such that τ(xw) = cτ(x) for x ∈ A. Since wτ is the canonical extension of τ(· w) = wτ on A to D_{ρ} and the canonical extension is unique, one has τ(xw) = cτ(x) for x ∈ D_{ρ} so that

τ(xw) = τ(xcp_{τ})  for x ∈ D_{ρ}.  (4.12)

As c = cτ(1) = τ(w), one has

1 = τ(p_{τ}) = τ(w^{∗}w) = cτ(w^{∗}) = c\overline{τ(w)} = cc̄

so that

(cp_{τ})^{∗}(cp_{τ}) = p_{τ} = w^{∗}w.

By the uniqueness of the polar decomposition, we have by (4.12) w = cp_{τ}. □

Proposition 4.11. Suppose that (A, ρ, Σ) is irreducible and satisfies dim E_{r_{ρ}}(ρ) = 1. Then dim E_{β}(ρ) ≤ 1 for β ∈ C with |β| = r_{ρ} > 1.

Proof. Let |β| = r_{ρ} > 1. Take an arbitrary linear functional φ ∈ E_{β}(ρ) with φ ≠ 0. Its canonical extension φ̃ to D_{ρ} is continuous. Denote by φ̃ = vφ̃|φ̃| the polar decomposition in D_{ρ}^{∗} where vφ̃ is a partial isometry in D_{ρ}^{∗∗}. By Lemma 4.7, the restriction of |φ̃| to A is a positive linear functional belonging to E_{r_{ρ}}(ρ). Since (A, ρ, Σ) is uniquely ergodic, by putting cφ̃ = |φ̃|(1) one has |φ̃| = cφ̃τ as a positive linear functional on A. The canonical extension to D_{ρ} which satisfies (4.3) is unique and determined by its behavior on A. Hence the equalty |φ̃| = cφ̃τ holds as a positive linear functional on D_{ρ} so that we have supp(|φ̃|) = supp(τ) and hence v∗ ; φ̃vφ̃ = p_{τ}. For another linear functional ψ ∈ E_{β}(ρ) with ψ ≠ 0, we have similar decompositions

ψ̃ = vψ̃|ψ̃|,  |ψ̃| = cψ̃τ,  v∗ ; ψ̃vψ̃ = p_{τ} .

Put a partial isometry w = v∗ ; φ̃vψ̃ ∈ D_{ρ}^{∗∗} so that w^{∗}w = p_{τ}. By Lemma 4.8, one has ϕ_{ρ}(w) = w. Lemma 4.10 implies w = cp_{τ} for some c ∈ C with |c| = 1 so that vψ̃ = cvφ̃. Therefore we have

ψ̃ = vψ̃|ψ̃| = cvφ̃cψ̃τ = c\frac{cψ̃}{cφ̃} φ̃

on D_{ρ}. In particular we have ψ = c\frac{cψ̃}{cφ̃} φ on A so that dim E_{β}(ρ) ≤ 1. □

Corollary 4.12. Suppose that (A, ρ, Σ) is irreducible and mean ergodic. Then for β ∈ C with |β| > 1, we have dim E_{β}(ρ) ≤ 1 if |β| = r_{ρ}, otherwise E_{β}(ρ) = {0}.

Suppose that (A, ρ, Σ) is irreducible and mean ergodic. Hence (A, ρ, Σ) is uniquely ergodic with a unique faithful invariant state τ ∈ E_{r_{ρ}}(ρ). Denote by p_{τ} ∈ D_{ρ}^{∗∗} the support projection of the canonical extension of τ to D_{ρ}, where the extension is still denoted by τ. For β = re^{iθ} ∈ C with r = r_{ρ} > 1, we set

P_{β}(D_{ρ}, τ) = {v ∈ D_{ρ}^{∗∗} | ϕ_{ρ}(v) = e^{iθ}v, v^{∗}v = p_{τ} }.

Denote by R_{+} the set of all nonnegative real numbers. For φ ∈ E_{β}(ρ) denote by φ̃ its canonical extension to D_{ρ}. As |β| = r_{ρ}, φ̃ is continuous and has a unique polar decomposition φ̃ = vφ̃|φ̃| for some vφ̃ ∈ D_{ρ}^{∗∗} and positive linear functional |φ̃| ∈ D_{ρ}^{∗}. By Lemma 4.8, we know the structure of the eigenspace E_{β}(ρ) as in the following way:

Proposition 4.13. Suppose that (A, ρ, Σ) is irreducible and mean ergodic. There exists a bijective correspondence between the eigenspace E_{β}(ρ) and the product set P_{β}(D_{ρ}, τ) × R_{+} through the correspondences

φ ∈E_{β}(ρ) ⟶ (vφ̃, |φ̃|(1)) ∈ P_{β}(D_{ρ}, τ) × R_{+},
cτ(· v) ∈E_{β}(ρ) ⟵ (v, c) ∈ P_{β}(D_{ρ}, τ) × R_{+}.

5. Extension to O_{ρ} and KMS condition

In [9], Enomoto-Fujii-Watatani have proved that KMS states for gauge action on the Cuntz-Krieger algebra O_{A} exist if and only if its inverse temperature is log r_{A}, where r_{A} is the Perron-Frobenius eigenvalue for the irreducible matrix A. They have showed that the KMS states bijectively correspond to the normalized positive eigenvectors of A for the eigenvalue r_{A}. In this section, we will study KMS conditions for linear functionals without assuming its positivity at inverse temperature taking complex numbers. The extended notation is needed to study eigenvector spaces for C^{∗}-symbolic dynamical systems. Following after [3], KMS states for one-parameter group action α on a C^{∗}- algebra B is defined as follows: For a positive real number γ ∈ R, a state ψ on B is a KMS state at inverse temperature γ if ψ satisfies

ψ(yα_{iγ}(x)) = ψ(xy),  x ∈ B^{a}, y ∈ B  (5.1)

where B^{a} is the set of analytic elements of the action α : R ⟶ Aut(B) (cf.[3]). The equation (5.1) for ψ is called the KMS condition with respect to the action α. In what follows, we restrict our interest to periodic actions so as to extend KMS condition to (not necessarily positive) linear functionals at inverse temperature taking complex numbers. We assume that an action α of R has its period 2π so that α is regarded as an action of one-dimensional torus group T = R/2πZ. Let B be a C^{∗}-algebra and α : T ⟶ Aut(B) a continuous action of T to the automorphism group Aut(B). We write a complex number β ∈ C as β = re^{iθ} where r, θ ∈ R with r > 1.

Definition. A continuous linear functional φ ∈ B^{∗} on B is said to satisfy KMS condition at Logβ if φ satisfies the following condition

φ(yα_{i log r}(x)) = φ(α_{θ}(x)y),  x ∈ B^{a}, y ∈ B.  (5.2)

Remark.

(i) As α_{θ}(x) = α_{θ+2π}(x), the right hand side φ(α_{θ}(x)y) of (5.2) does not depend on the choice of θ ∈ R as long as β = re^{iθ}.

(ii) The above KMS condition (5.2) is equivalent to the following condition:

φ(yα_{ζ+i log r}(x)) = φ(α_{ζ+θ}(x)y),  x ∈ B^{a}, y ∈ B, ζ ∈ C (5.3)

(iii) In case of θ = 0, the above definition of KMS condition coincides with

the original definition of KMS condition for states. (iv) The above equality (5.2) can be written formally as

φ(yα_{iLogβ}(x)) = φ(xy),  x ∈ B^{a}, y ∈ B,  (5.4)

if we denote Logβ = log r + iθ. We will present some examples of linear functionals satisfying the extended KMS conditions. Examples.

(i) Let α : T ⟶ Aut(B) be an action of T to a C^{∗}-algebra B such that there exists a projection H ∈ B satisfying α_{t}(a) = e^{itH}ae^{−itH}, a ∈ B, t ∈ T. Assume that there exists an α-invariant tracial state tr on B. Put

φ(x) = \frac{tr(e^{−LogβH}x)}{tr(e^{−LogβH})}, x ∈ B,

where Logβ = log r + iθ. Then φ satisfies KMS condition at Logβ. (ii) Let B = ⊗_{k=1}^{∞}M_{2} be the UHF-algebra of type 2^{∞} and α : T ⟶ Aut(B)

an action of T to B defined by

α_{t} = ⊗_{k=1}^{∞}Ad [\begin{matrix}1 & 0 \\ 0 & e^{it} \end{matrix}] , t ∈ T.

Put

B_{n} = ⊗_{k=1}^{n} M_{2} = M_{2} ⊗ ⋯ ⊗ M_{2},
u_{t}^{n} = ⊗_{k=1}^{n} [\begin{matrix}1 & 0 \\ 0 & e^{it} \end{matrix}] = [\begin{matrix}1 & 0 \\ 0 & e^{it} \end{matrix}] ⊗ ⋯ ⊗ [\begin{matrix}1 & 0 \\ 0 & e^{it} \end{matrix}] ∈ B_{n},
α_{t}^{n} =Ad(u_{t}^{n}) ∈ Aut(B_{n}), t ∈ T.

Let β = re^{iθ} ∈ C be r > 1. Put

H = [\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}] ∈ M_{2}, h_{n} = ⊗_{k=1}^{n} [\begin{matrix}1 & 0 \\ 0 & \frac{1}{β} \end{matrix}] ∈ B_{n},

and hence h_{n} = ⊗_{k=1}^{n}e^{−LogβH}, α_{t}^{n} = ⊗_{k=1}^{n}Ad(e^{itH}), t ∈ T. It is straightforward to see that

tr(e^{−LogβH}bα_{i log r}(a)) = tr(e^{−LogβH}α_{θ}(a)b), a, b ∈ M_{2}.

Put

φ_{n}(x) = ⊗_{k=1}^{n}tr(xh_{n}) for x ∈ B_{n}

so that we have

φ_{n}(yα_{i log r}(x)) = φ_{n}(α_{θ}(x)y), x, y ∈ B_{n}.

As ∥h_{n}∥ = 1, φ_{n} extends to a continuous linear functional on B, which we denote by φ. Then φ satisfies KMS condition at Logβ:

φ(yα_{i log r}(x)) = φ_{n}(α_{θ}(x)y), x ∈ B^{a}, y ∈ B.

We see the following two propositions whose proofs are similar to the case of usual KMS states.

Proposition 5.1 (cf. [39, 8.12.3]). Let α : T ⟶ Aut(B) be a continuous action of T to the automorphism group Aut(B) of a C^{∗}-algebra B and β a complex number with β = re^{iθ}, r > 1. The following conditions for a continuous linear functional φ on B are equivalent:

(i) φ satisfies the KMS condition at Logβ.

(ii) φ satisfies the equality (5.2) for just a dense set of elements in B^{a}. (iii) For all x, y ∈ B, there is a bounded continuous function f on the strip

Ω_{log r} = {ζ ∈ C | 0 ≤ Imζ ≤ log r}

such that f is holomorphic in the interior of Ω_{log r} and

f(t) = φ(yα_{t}(x)),  f(t + i log r) = φ(α_{t+θ}(x)y),  t ∈ R.

Proposition 5.2 (cf. [39, 8.12.4]). Let B be a C^{∗}-algebra and α : T ⟶ Aut(B) be a continuous action of T to the automorphism group Aut(B). Let φ be a continuous linear functional on B. If φ satisfies KMS condition at Logβ for some complex number β with β = re^{iθ} with r > 1, then φ is α-invariant, that is,

φ ∘ α_{t} = φ, t ∈ T.

We henceforth go back to our previous situations. Let (A, ρ, Σ) be a C^{∗}- symbolic dynamical system. Recall that the positive operator λ_{ρ} on A extends to F_{ρ} by setting λ_{ρ}(x) = ∑_{α∈Σ} S_{α}^{∗}xS_{α}, x ∈ F_{ρ}. For β ∈ C with β ≠ 0, we set

E_{β}^{D}(ρ) = {φ ∈ D_{ρ}^{∗} | φ(λ_{ρ}(x)) = βφ(x), x ∈ D_{ρ}},  (5.5)
E_{β}^{F}(ρ) = {ϕ ∈ F_{ρ}^{∗} | ϕ(λ_{ρ}(x)) = βϕ(x), x ∈ F_{ρ}, ϕ is tracial on F_{ρ}}. (5.6)

It is possible that both E_{β}^{D}(ρ) and E_{β}^{F}(ρ) are {0}. Recall that E_{D} : F_{ρ} ⟶ D_{ρ} is the canonical expectation satisfying by E_{D}(S_{μ}aS_{ν}^{∗}) = δ_{μ,ν}S_{μ}aS_{ν}^{∗} for a ∈ A with μ, ν ∈ B_{∗}(Λ), |μ| = |ν|. By composing it to a given linear functional φ ∈ E_{β}^{D}(ρ) on D_{ρ}, φ extends to F_{ρ}.

Lemma 5.3. Let β ∈ C with |β| > 1. A (not necessarily positive) continuous linear functional φ ∈ E_{β}^{D}(ρ) on D_{ρ} uniquely extends to F_{ρ} as a tracial continuous linear functional ϕ = φ ∘ E_{D} such that

ϕ(S_{μ}xS_{ν}^{∗}) = δ_{μ,ν} \frac{1}{β^{|μ|}} ϕ(xS_{μ}^{∗}S_{ν}), x ∈ F_{ρ}, μ, ν ∈ B_{∗}(Λ) with |μ| = |ν|. (5.7)

Hence the sets E_{β}^{D}(ρ) and E_{β}^{F}(ρ) bijectively correspond to each other.

Proof. For φ ∈ E_{β}^{D}(ρ), as in the proof of Lemma 4.3 (i) ⇒ (ii), the equality

φ(S_{μ}aS_{μ}^{∗}) = \frac{1}{β^{|μ|}} φ(aρ_{μ}(1)), a ∈ A, μ ∈ B_{∗}(Λ)

holds so that

ϕ(S_{μ}aS_{ν}^{∗}) = δ_{μ,ν} \frac{1}{β^{|μ|}} φ(aρ_{μ}(1)), a ∈ A, μ, ν ∈ B_{∗}(Λ) with |μ| = |ν|.

By Lemma 4.3 (iii) ⇒ (i), ϕ belongs to E_{β}^{F}(ρ). □

Recall that E_{ρ} : O_{ρ} ⟶ O_{ρ}ρ̂ = F_{ρ} denotes the conditional expectation defined by (2.3).

Proposition 5.4. For any tracial continuous linear functional ϕ ∈ E_{β}^{F}(ρ), the composition ψ = ϕ ∘ E_{ρ} is a continuous linear functional on O_{ρ} which satisfies KMS condition at Logβ for gauge action ρ̂ of T.

Proof. Let P_{ρ} be the dense ∗-subalgebra of O_{ρ} generated algebraically by S_{α}, α ∈ Σ and a ∈ A. It is clear that for each element x ∈ P_{ρ} the function t ∈ T = R/2πR → ρ̂_{t}(x) ∈ O_{ρ} extends to an entire analytic function on C. Put ψ = ϕ ∘ E_{ρ}. We will show that the equality (5.2) holds for ψ. Elements x, y ∈ P_{ρ} can be expanded as finite linear combinations

x = ∑ x_{−ν}S_{ν}^{∗} + x_{0} + ∑ S_{μ}x_{μ}, y = ∑ y_{−ν}S_{ν}^{∗} + y_{0} + ∑ S_{μ}y_{μ} (5.8)

for some x_{−ν}, x_{0}, x_{μ}, y_{−ν}, y_{0}, y_{μ} ∈ F_{ρ}^{alg}. As ψ is a tracial linear functional on F_{ρ}, it suffices to check the equality (5.2) for the following two cases

(1) x = S_{ν}x_{ν}, y = y_{−ν}S_{ν}^{∗}, (2) x = x_{−μ}S_{μ}^{∗}, y = S_{μ}y_{μ}.

Case (1):

ψ(yρ̂_{i log r}(x)) = ψ(y_{−ν}S_{ν}^{∗}e^{−|ν| log r}S_{ν}x_{ν})

= \frac{1}{β^{|ν|}} ψ(e^{i|ν|θ}x_{ν}y_{−ν}S_{ν}^{∗}S_{ν})

= ψ(e^{i|ν|θ}S_{ν}x_{ν}y_{−ν}S_{ν}^{∗})

= ψ(ρ̂_{θ}(x)y).

Case (2):

ψ(yρ̂_{i log r}(x)) = ψ(S_{μ}y_{μ}e^{|μ| log r}x_{−μ}S_{μ}^{∗})

= \frac{r^{|μ|}}{β^{|μ|}} ψ(y_{μ}x_{−μ}S_{μ}^{∗}S_{μ})

= ψ(e^{−i|μ|θ}x_{−μ}S_{μ}^{∗}S_{μ}y_{μ})

= ψ(ρ̂_{θ}(x)y).

This completes the proof. □

Conversely we have

Lemma 5.5. If a continuous linear functional ψ on O_{ρ} satisfies KMS condition at Logβ for some β ∈ C with |β| > 1, then the restriction ϕ = ψ|_{F_{ρ}} to F_{ρ} belongs to E_{β}^{F}(ρ) and satisfies the equality ψ = ϕ ∘ E_{ρ}.

Proof. Let β = re^{iθ} with r > 1. For any x ∈ F_{ρ}, μ ∈ B_{∗}(Λ), we see

ψ(S_{μ}x) = \frac{1}{β^{|μ|}} ψ(xS_{μ}) = \frac{1}{β^{|μ|}} ψ(S_{μ}ρ̂_{i log r}(α_{−θ}(x))) = \frac{1}{β^{|μ|}} ψ(S_{μ}x)

so that ψ(S_{μ}x) = 0 because |β| > 1. We similarly have ψ(xS_{μ}^{∗}) = 0. Since any element of P_{ρ} can be expanded as in (5.8), we get ψ(y) = ϕ ∘ E_{ρ}(y) for y ∈ P_{ρ}. We will next show that ϕ belongs to E_{β}^{F}(ρ). For x, y ∈ F_{ρ}, one sees ρ̂_{i log r}(x) = ρ̂_{−θ}(x) = x so that ψ(yx) = ψ(xy). Hence ψ gives rise to a tracial linear functional ϕ on F_{ρ}. By KMS condition, we get for any x ∈ F_{ρ}, μ ∈ B_{∗}(Λ),

ψ(S_{μ} · xS_{μ}^{∗}) = ψ(xS_{μ}^{∗}ρ̂_{i log r}(ρ̂_{−θ}(S_{μ})) = \frac{1}{β^{|μ|}} ψ(xS_{μ}^{∗}S_{μ}).

Thus by Lemma 4.3, we know ϕ ∈ E_{β}^{F}(ρ). □

We set for β ∈ C with |β| > 1,

KMS_{β}(O_{ρ})
={ψ ∈ O_{ρ}^{∗} | ψ satisfies KMS condition at Logβ for gauge action}

and

Sp(ρ) = {β ∈ C | φ ∘ λ_{ρ} = βφ for some φ ∈ A^{∗} with φ ≠ 0}.

By Proposition 5.4 and Lemma 5.5, we have

Proposition 5.6. Let (A, ρ, Σ) be an irreducible C^{∗}-symbolic dynamical system. Assume that (A, ρ, Σ) is power-bounded. Let β ∈ C be a complex number with |β| > 1. If |β| = r_{ρ} and β ∈ Sp(ρ), we have KMS_{β}(O_{ρ}) ≠ {0}. If in particular, (A, ρ, Σ) is mean ergodic, KMS_{β}(O_{ρ}) ≠ {0} if and only if |β| = r_{ρ} and β ∈ Sp(ρ).

Proof. Under the assumption that (A, ρ, Σ) is power-bounded, any continuous linear functional φ ∈ E_{β}(ρ) on A can uniquely extend to a continuous linear functional φ̃ on D_{ρ}, that belongs to E_{β}^{D}(ρ) if |β| = r_{ρ}. By Proposition 5.4, φ̃ ∘ E_{D} ∈ E_{β}^{F}(ρ) has an extension on O_{ρ} as a continuous linear functional that satisfies KMS condition at Logβ. Conversely, the restriction of a continuous linear functional KMS_{β}(O_{ρ}) to the subalgebra A yields a nonzero element of E_{β}(ρ) which has continuous extension to D_{ρ}. If in particular, (A, ρ, Σ) is mean ergodic, |β| must be r_{ρ} by Theorem 4.9. □

Therefore we conclude

Theorem 5.7. Let (A, ρ, Σ) be an irreducible C^{∗}-symbolic dynamical system. Let β ∈ C be a complex number with |β| = r_{ρ} > 1.

(i) Suppose that (A, ρ, Σ) is power-bounded. Then there exist linear isomorphisms among the four spaces E_{β}(ρ), E_{β}^{D}(ρ), E_{β}^{F}(ρ) and KMS_{β}(O_{ρ}) through the correspondences φ ∈ E_{β}(ρ), φ̃ ∈ E_{β}^{D}(ρ), φ̃ ∘ E_{D} ∈ E_{β}^{F}(ρ), φ̃ ∘ E_{D} ∘ E_{ρ} ∈ KMS_{β}(O_{ρ}) respectively. In particular, there exists a bijective correspondence between the set E_{β}(ρ) of eigenvectors of λ_{ρ}^{∗} for eigenvalue β consisting of continuous linear functionals on A and the set KMS_{β}(O_{ρ}) of continuous linear functionals on O_{ρ} satisfying KMS condition at Logβ.

(ii) Suppose that (A, ρ, Σ) is mean ergodic. Then the dimension dim KMS_{β}(O_{ρ}) of the space of continuous linear functionals on O_{ρ} satisfying KMS condition at Logβ is one if there exists a nonzero eigenvector of λ_{ρ}^{∗} on A^{∗} for the eigenvalue β. In particular there uniquely exists a faithful KMS state on O_{ρ} at log r_{ρ}.

The following corollary is a generalization of [9, Theorem 6].

Corollary 5.8. Suppose that A is an irreducible matrix with entries in {0, 1} with its period p_{A}. Let β be a complex number with |β| > 1.

(i) There exists a nonzero continuous linear functional on the Cuntz-Krieger algebra O_{A} satisfying KMS condition for gauge action at Logβ if and only if β is a p_{A}-th root of the Perron-Frobenius eigenvalue r_{A} of A.

(ii) The space of admitted continuous linear functionals on O_{A} satisfying

KMS condition for gauge action at Logβ is of one-dimensional. (iii) If in particular β = r_{A}, the space of admitted continuous linear func-

tionals on O_{A} satisfying KMS condition for gauge action at log r_{A} is the scalar multiples of a unique KMS state.

6. KMS states and invariant measures

In this section, we will study a relationship between KMS states on O_{ρ} and invariant measures on D_{ρ} under ϕ_{ρ}. In what follows we assume that (A, ρ, Σ) is irreducible and fix a faithful invariant state τ on A.

We denote by ∥a∥_{2} the L^{2}-norm τ(a^{∗}a)^{\frac{1}{2}} for a ∈ A, and by H_{τ} the completion of A by the norm ∥ · ∥_{2}. By the inequalities for n ∈ N, a ∈ A

τ(λ_{ρ}^{n}(a)^{∗}λ_{ρ}^{n}(a)) ≤ ∥λ_{ρ}^{n}∥τ(λ_{ρ}^{n}(a^{∗}a)) = ∥λ_{ρ}^{n}∥r_{ρ}^{n}τ(a^{∗}a) ≤ ∥λ_{ρ}^{n}∥^{2}∥a∥_{2}^{2}, (6.1)

the operators T_{ρ}^{n}, n ∈ N induce bounded linear operators on H_{τ}. The induced operators on H_{τ}, which we also denote by T_{ρ}^{n}, n ∈ N, are uniformly bounded in the operator norm on H_{τ}, if (A, ρ, Σ) is power-bonded. We provide the following lemma, which shows power-boundedness of (A, ρ, Σ) induces an ordinary mean ergodicity on H_{τ}, is a direct consequence from [22, p.73,Theorem 1.2]. We give a proof for the sake of completeness.

Lemma 6.1. Suppose that (A, ρ, Σ) is irreducible and power-bounded. Then

lim_{n→∞} \frac{1 + T_{ρ} + T_{ρ}^{2} + ⋯ + T_{ρ}^{n−1}}{n}

converges to an idempotent P_{ρ} on H_{τ} under strong operator topology in B(H_{τ}). The subspace P_{ρ}H_{τ} consists of the vectors of H_{τ} fixed under T_{ρ}.

Proof. The mean operators M_{n}, n ∈ N on A defined by (3.1) naturally act on H_{τ}. Since (A, ρ, Σ) is power-bounded, there exists a positive number c > 0 such that ∥T_{ρ}^{n}∥ < c for all n ∈ N. As ∥M_{n}∥ < 1 + c, n ∈ N, the sequence M_{n}v ∈ H_{τ} , n ∈ N for a vector v ∈ H_{τ} has a cluster point v_{0} under the weak topology of H_{τ}. The identites

(I − T_{ρ})M_{n} = M_{n}(I − T_{ρ}) = \frac{1}{n}(I − T_{ρ}^{n})

imply the inequalites

∥(I − T_{ρ})M_{n}∥ = ∥M_{n}(I − T_{ρ})∥ = \frac{1}{n}∥I − T_{ρ}^{n}∥ < \frac{1}{n}(1 + c).  (6.2)

Hence we have T_{ρ}v_{0} = v_{0}. Put

Q_{n} = \frac{1}{n}\{(I + T_{ρ}) + (I + T_{ρ} + T_{ρ}^{2}) + ⋯ + (I + T_{ρ} + ⋯ + T_{ρ}^{n−2})\}.

Then we have v − M_{n}v = (I − T_{ρ})Q_{n}v, n ∈ N. Hence v − v_{0} belongs to the weak closure K_{τ} of the subspace (I − T_{ρ})H_{τ}. The weak closure K_{τ} is also the norm closure of the subspace (I − T_{ρ})H_{τ}. For w ∈ K_{τ}, take w_{j} ∈ (I − T_{ρ})H_{τ} such that ∥w − w_{j}∥_{2} → 0 and w_{j} = (I − T_{ρ})x_{j} for some x_{j} ∈ H_{τ}. We then have by (6.2)

∥M_{n}w∥_{2} ≤ ∥M_{n}∥∥w − w_{j}∥_{2} + ∥M_{n}(I − T_{ρ})x_{j}∥_{2}

≤ (1 + c)∥w − w_{j}∥_{2} + \frac{1}{n}(1 + c)∥x_{j}∥_{2}

so that lim_{n→∞} ∥M_{n}w∥_{2} = 0. Since M_{n}v − v_{0} = M_{n}(v − v_{0}) and v − v_{0} ∈ K_{τ}, one has

lim_{n→∞} ∥M_{n}v − v_{0}∥_{2} = 0.

Put P_{ρ}v = v_{0}. The inequality

∥M_{n}v − T_{ρ}M_{n}v∥_{2} = ∥(I − T_{ρ})M_{n}v∥_{2} < \frac{1}{n}(1 + c)∥v∥_{2}

implies that P_{ρ} = T_{ρ}P_{ρ} that is equal to P_{ρ}T_{ρ}. Therefore P_{ρ} = M_{n}P_{ρ} = P_{ρ}M_{n} and hence P_{ρ} = P_{ρ}^{2}. □

Remark. Under the same assumption above, one may prove that the limit

lim_{r↓r_{ρ}}(r − r_{ρ})R(r)

for the resolvent R(r) = (r − λ_{ρ})^{−1} with r > r_{ρ} converges to the idempotent P_{ρ} on H_{τ} under strong operator topology in B(H_{τ}). Hence the equality

lim_{r↓r_{ρ}}(r − r_{ρ})R(r) = lim_{n→∞} \frac{1 + T_{ρ} + T_{ρ}^{2} + ⋯ + T_{ρ}^{n−1}}{n}  (6.3)

holds. We will give a proof of the equality (6.3). It is enough to consider the limit lim_{n→∞} \frac{1}{n}R(r_{ρ} + \frac{1}{n}) instead of lim_{r↓r_{ρ}}(r − r_{ρ})R(r). As in the above proof, there exists c > 0 such that ∥T_{ρ}^{k}(a)∥_{2} ≤ c∥a∥_{2} for a ∈ A, k ∈ N. Put R_{n} = \frac{1}{n}R(r_{ρ} + \frac{1}{n}). Since for y ∈ A

R(r_{ρ} + \frac{1}{n})y = ∑_{k=0}^{∞} \frac{λ_{ρ}^{k}(y)}{(r_{ρ} + \frac{1}{n})^{k+1}}

one has

∥R(r_{ρ} + \frac{1}{n})y∥_{2} ≤ ∑_{k=0}^{∞} ∥T_{ρ}^{k}(y)∥\frac{r_{ρ}^{k}}{(r_{ρ} + \frac{1}{n})^{k+1}} ≤ nc∥y∥_{2}

and hence ∥R_{n}∥ ≤ c for n ∈ N. The identites

(I − T_{ρ})R_{n} = R_{n}(I − T_{ρ}) = \frac{1}{n} \frac{1}{r_{ρ}} (R_{n} − I)

hold so that we have

∥(I − T_{ρ})R_{n}∥ = ∥R_{n}(I − T_{ρ})∥ ≤ \frac{1}{n} \frac{1}{r_{ρ}} (1 + c).

A similar argument to the proof of Lemma 6.1 works so that for u ∈ H_{τ} by taking a cluster point u_{0} of the sequence R_{n}u, n ∈ ℕ under the weak topology of H_{τ} we have

lim_{n→∞} ∥R_{n}u − u_{0}∥_{2} = 0.

Put P̂_{ρ}u = u_{0}. The inequality ∥R_{n}u − T_{ρ}R_{n}u∥_{2} ≤ \frac{1}{n} \frac{1}{r_{ρ}} (1 + c)∥u∥_{2} implies that

P̂_{ρ} = T_{ρ}P̂_{ρ} that is equal to P̂_{ρ}T_{ρ}. Hence P̂_{ρ} = R_{n}P̂_{ρ} and P̂_{ρ} = P̂_{ρ}^{2}. The equality P̂_{ρ} = T_{ρ}P̂_{ρ} implies P̂_{ρ} = M_{n}P̂_{ρ} for all n ∈ N so that P̂_{ρ} = P_{ρ}P̂_{ρ}. Similarly the

equalities P_{ρ} = T_{ρ}P_{ρ} and R_{n} = ∑_{k=0}^{∞} T_{ρ}^{k} \frac{r_{ρ}^{k}}{(r_{ρ}+ \frac{1}{n})^{k+1}} imply P_{ρ} = R_{n}P_{ρ} for all

n ∈ N so that P_{ρ} = P̂_{ρ}P_{ρ}. As P_{ρ}P̂_{ρ} = P̂_{ρ}P_{ρ}, one has P_{ρ} = P̂_{ρ}.

We denote by ∥a∥_{1} the L^{1}-norm τ(|a|) of a ∈ A, and by L^{1}(A, τ) the completion of A by the norm ∥ · ∥_{1}. The positive operators λ_{ρ}, T_{ρ} : A ⟶ A and the state τ : A ⟶ C extend to L^{1}(A, τ) in natural way, that are also denoted by λ_{ρ}, T_{ρ} and τ respectively.

Lemma 6.2. Suppose that (A, ρ, Σ) is uniquely ergodic and power-bounded. Then for a ∈ A the limit lim_{n→∞} M_{n}(a) converges in L^{1}(A, τ) under ∥ · ∥_{1}- topology. In particular lim_{n→∞} M_{n}(1) = x_{ρ} exists in L^{1}(A, τ) and satisfies the equalities

τ(x_{ρ}) = 1 and lim_{n→∞} M_{n}(a) = τ(a)x_{ρ}  for a ∈ A.  (6.4)

Proof. Since (A, ρ, Σ) is irreducible and power-bounded, lim_{n→∞} M_{n}(a) for a ∈ A converges in H_{τ} = L^{2}(A, τ) under ∥ · ∥_{2}-norm by the previous lemma. By the inequality

∥M_{n}(a) − M_{m}(a)∥_{1} ≤ ∥M_{n}(a) − M_{m}(a)∥_{2}, a ∈ A

the limit lim_{n→∞} M_{n}(a) exists in L^{1}(A, τ) under ∥ · ∥_{1}-norm. We denote it by Φ_{1}(a). Hence x_{ρ} = Φ_{1}(1). We will show that τ(f(Φ_{1}(a) − τ(a)x_{ρ})) = 0 for f ∈ A. It suffices to show that τ(bΦ_{1}(a)b^{∗}) = τ(a)τ(bx_{ρ}b^{∗}) for b ∈ A. One may assume that a ≥ 0. The inequality a ≤ ∥a∥1 and hence M_{n}(a) ≤ ∥a∥M_{n}(1) implies bΦ_{1}(a)b^{∗} ≤ ∥a∥bx_{ρ}b^{∗} so that we have 0 ≤ τ(bΦ_{1}(a)b^{∗}) ≤ ∥a∥τ(bx_{ρ}b^{∗}). Hence τ(bx_{ρ}b^{∗}) = 0 implies τ(bΦ_{1}(a)b^{∗}) = 0. We may assume that τ(bx_{ρ}b^{∗}) ≠ 0. Put ω(a) = \frac{τ(bΦ_{1}(a)b^{∗})}{τ(bx_{ρ}b^{∗})}, a ∈ A. As Φ_{1} ∘ T_{ρ}(a) = Φ_{1}(a), one sees that ω is an invariant state on A. Hence we have ω = τ by the unique ergodicity of (A, ρ, Σ). Therefore we have τ(bΦ_{1}(a)b^{∗}) = τ(a)τ(bx_{ρ}b^{∗}) for b ∈ A. The equality τ(x_{ρ}) = 1 is clear. □

Lemma 6.3. Keep the above assumptions and notations. The limit lim_{n→∞} M_{n}(f) for f ∈ L^{1}(A, τ) converges in L^{1}(A, τ) under ∥ · ∥_{1}-topology and satisfies the equality

lim_{n→∞} M_{n}(f) = τ(f)x_{ρ} for f ∈ L^{1}(A, τ).

Proof. Since for f ∈ L^{1}(A, τ) the inequality |λ_{ρ}(f)| ≤ λ_{ρ}(|f|) holds, one has |T_{ρ}(f)| ≤ T_{ρ}(|f|) and hence ∥M_{n}(f)∥_{1} ≤ ∥f∥_{1}. Take a_{k} ∈ A such as ∥f − a_{k}∥_{1} → 0 as k → ∞. It then follows that

∥M_{n}(f) − τ(f)x_{ρ}∥_{1}
≤∥M_{n}(f) − M_{n}(a_{k})∥_{1} + ∥M_{n}(a_{k}) − τ(a_{k})x_{ρ}∥_{1} + ∥τ(a_{k})x_{ρ} − τ(f)x_{ρ}∥_{1}
≤∥f − a_{k}∥_{1} + ∥M_{n}(a_{k}) − τ(a_{k})x_{ρ}∥_{1} + |τ(a_{k}) − τ(f)|∥x_{ρ}∥_{1},

and hence lim_{n→∞} ∥M_{n}(f) − τ(f)x_{ρ}∥_{1} = 0 by the preceding lemma. □

Proposition 6.4. Keep the above assumptions and notations. If f ∈ L^{1}(A, τ) satisfies T_{ρ}(f) = f and τ(f) = 1, Then f = x_{ρ}. Namely the space of the fixed elements in L^{1}(A, τ) under T_{ρ} is one-dimensional.

Proof. By the preceding lemma, we have for f ∈ L^{1}(A, τ) lim_{n→∞} M_{n}(f) = τ(f)x_{ρ} in ∥ · ∥_{1}-topology. By the condition T_{ρ}(f) = f, we have M_{n}(f) = f with τ(f) = 1 and hence f = x_{ρ}. □

Let us define the space L^{1}(D_{ρ}, τ) in a similar way to L^{1}(A, τ). The operators λ_{ρ}, T_{ρ} : D_{ρ} ⟶ D_{ρ} and the state τ : D_{ρ} ⟶ C naturally act on L^{1}(D_{ρ}, τ). The inclusion relation A ⊂ D_{ρ} induces the inclusion relation L^{1}(A, τ) ⊂ L^{1}(D_{ρ}, τ).

Lemma 6.5. Keep the above assumptions and notations. Let x be an element of L^{1}(D_{ρ}, τ) such that T_{ρ}(x) = x. Then x belongs to L^{1}(A, τ).

Proof. Take x_{n} ∈ D_{ρ}^{alg} such that ∥x_{n} − x∥_{1} → 0 as n → ∞. As |λ_{ρ}(y)| ≤ λ_{ρ}(|y|), y ∈ D_{ρ}, it then follows that

∥λ_{ρ}(x_{n}) − λ_{ρ}(x)∥_{1} = τ(|λ_{ρ}(x_{n} − x)|) ≤ τ(λ_{ρ}(|x_{n} − x|) = r_{ρ}∥x_{n} − x∥_{1}

so that ∥T_{ρ}(x_{n}) − T_{ρ}(x)∥_{1} ≤ ∥x_{n} − x∥_{1}. The element x is fixed by T_{ρ} so that

∥T_{ρ}^{k}(x_{n}) − x∥_{1} ≤ ∥x_{n} − x∥_{1}, n ∈ N, k ∈ N.

Since x_{n} ∈ D_{ρ}^{alg}, there exists k_{n} ∈ N such that T_{ρ}^{k_{n}}(x_{n}) ∈ A. Hence x belongs to L^{1}(A, τ). □

Definition. A state μ on D_{ρ} is called a ϕ_{ρ}-invariant measure if it satisfies

μ(y) = μ(ϕ_{ρ}(y)), y ∈ D_{ρ}.

If the probability measure for a state μ on D_{ρ} is absolutely continuous with respect to the probability measure for the state τ on D_{ρ}, we write it as μ ≪ τ.

Proposition 6.6. Assume that (A, ρ, Σ) is irreducible and uniquely ergodic. For a fixed positive element x ∈ L^{1}(A, τ) by T_{ρ} satisfying τ(x) = 1, the state μ_{x} on D_{ρ} defined by

μ_{x}(y) = τ(yx), y ∈ D_{ρ}

is a ϕ_{ρ}-invariant measure on D_{ρ} such that μ ≪ τ. Conversely, for any ϕ_{ρ}- invariant measure μ on D_{ρ} such that μ ≪ τ, there exists a fixed positive element x_{μ} ∈ L^{1}(A, τ) by T_{ρ} satisfying τ(x_{μ}) = 1 such that

μ(y) = τ(yx_{μ}), y ∈ D_{ρ}.

Proof. Let x ∈ L^{1}(A, τ) be a fixed positive element by T_{ρ} satisfying τ(x) = 1. As λ_{ρ}(x) = r_{ρ}x, it follows that from Lemma 4.7

μ_{x}(ϕ_{ρ}(y)) = \frac{1}{r_{ρ}} τ(λ_{ρ}(ϕ_{ρ}(y)x)) = \frac{1}{r_{ρ}} τ(yλ_{ρ}(x)) = μ_{x}(y), y ∈ D_{ρ}

so that the state μ_{x} is a ϕ_{ρ}-invariant measure on D_{ρ} such that μ_{x} ≪ τ. Conversely for a ϕ_{ρ}-invariant measure μ on D_{ρ} such that μ ≪ τ, there exists a Radon-Nikodym derivative x_{μ} ∈ L^{1}(D_{ρ}, τ) such that x_{μ} ≥ 0, τ(x_{μ}) = 1 and

μ(y) = τ(yx_{μ}), y ∈ D_{ρ}.

By the equality τ(ϕ_{ρ}(y)x_{μ}) = τ(yT_{ρ}(x_{μ})), y ∈ D_{ρ}, one sees that τ(yx_{μ}) = τ(yT_{ρ}(x_{μ})), y ∈ D_{ρ} so that T_{ρ}(x_{μ}) = x_{μ}, τ − a.e. Hence x_{μ} is regarded as an element of L^{1}(A, τ) by the preceding lemma. This completes the proof. □

Especially the measure μ_{ρ} defined by μ_{ρ}(y) = τ(yx_{ρ}), y ∈ D_{ρ} is a ϕ_{ρ}-invariant measure on D_{ρ} such that μ_{ρ} ≪ τ. Therefore we have

Theorem 6.7. Assume that (A, ρ, Σ) is irreducible, uniquely ergodic and power-bounded. Then a ϕ_{ρ}-invariant measure on D_{ρ} absolutely continuous with respect to τ is unique and is of the form

μ_{ρ}(y) = τ(yx_{ρ}),  y ∈ D_{ρ}.  (6.5)

The measure μ_{ρ} is faithful, and ergodic in the sense that the formula

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} μ_{ρ}(ϕ_{ρ}^{k}(y)x) = μ_{ρ}(y)μ_{ρ}(x), x, y ∈ D_{ρ}

holds.

Proof. Let μ be a ϕ_{ρ}-invariant measure on D_{ρ}. By the preceding proposition there exists a fixed positive element x_{μ} ∈ L^{1}(A, τ) under T_{ρ} satisfying τ(x_{μ}) = 1 such that

μ(y) = τ(yx_{μ}), y ∈ D_{ρ}.

By Proposition 6.4 we have x_{μ} = x_{ρ}. For x, y ∈ D_{ρ}, the equality

λ_{ρ}^{k}(ϕ_{ρ}^{k}(y)xx_{ρ}) = yλ_{ρ}^{k}(xx_{ρ})

holds by Lemma 4.7 so that

\frac{1}{n} ∑_{k=0}^{n−1} μ_{ρ}(ϕ_{ρ}^{k}(y)x) = \frac{1}{n} ∑_{k=0}^{n−1} τ(ϕ_{ρ}^{k}(y)xx_{ρ})

= \frac{1}{n} ∑_{k=0}^{n−1} \frac{1}{r_{ρ}^{k}} τ(λ_{ρ}^{k}(ϕ_{ρ}^{k}(y)xx_{ρ}))

= \frac{1}{n} ∑_{k=0}^{n−1} \frac{1}{r_{ρ}^{k}} τ(yλ_{ρ}^{k}(xx_{ρ}))

= τ(yM_{n}(xx_{ρ})).

Since

∥ · ∥_{1} − lim_{n→∞} M_{n}(xx_{ρ}) = τ(xx_{ρ})x_{ρ} = μ_{ρ}(x)x_{ρ},

we have

lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} μ_{ρ}(ϕ_{ρ}^{k}(y)x) = τ(yμ_{ρ}(x)x_{ρ})) = μ_{ρ}(y)μ_{ρ}(x).

□

Corollary 6.8. Assume that (A, ρ, Σ) is irreducible and mean ergodic.

(i) The unique ϕ_{ρ}-invariant probability measure absolutely continuous with respect to τ is obtained by μ_{ρ}(y) = τ(yx_{ρ}), y ∈ D_{ρ}, where τ is the restriction of the unique KMS state on O_{ρ} and x_{ρ} is a positive element of A defined by the limit of the mean lim_{n→∞} \frac{1}{n}(1 + T_{ρ}(1) + ⋯ + T_{ρ}^{n−1}(1)).

(ii) The state μ_{ρ} is equivalent to the state τ as a measure on D_{ρ}.

Proof. (i) Under the assumption that (A, ρ, Σ) is irreducible. Mean ergodicity implies unique ergodicity and (FP), which implies power-boundedness. Therefore the assertion is immediate. (ii) By the mean ergodicity, the fixed element x_{ρ} belongs to A and is strictly positive by Lemma 3.5 (ii). Hence we have τ(y) = μ_{ρ}(yx_{ρ}^{−1}), y ∈ D_{ρ} so that τ ≪ μ_{ρ}. □

7. Examples

We will present examples of continuous linear functionals satisfying KMS conditions on some C^{∗}-symbolic dynamical systems.

1. Finite directed graphs Let A = [A(i, j)]_{i,j=1,...,N} be an N × N matrix with entries in nonnegative integers. Denote by G_{A} = (V_{A}, E_{A}) the associated finite directed graph with vertex set V = {v_{1}, …, v_{N}} and edge set E_{A}. Let O_{A^{[2]}} be the Cuntz-Krieger algebra such that the generating partial isometries S_{e}, e ∈ E_{A} indexed by the edges in G_{A} satisfy

∑_{f∈E_{A}} S_{f}S_{f}^{∗} = 1,  S_{e}^{∗}S_{e} = ∑_{f∈E_{A}} A^{[2]}(e, f)S_{f}S_{f}^{∗},  e ∈ E_{A},

where A^{[2]}(e, f) is defined to be one if the edge f follows the edge e, otherwise zero. Put A_{G_{A}} the C^{∗}-subalgebra of O_{A^{[2]}} generated by the projections S_{e}^{∗}S_{e}, e ∈ E_{A}. Denote by ρ_{e}^{A} for e ∈ E_{A} the endomorphism A_{G_{A}} defined by ρ_{e}^{A}(a) = S_{e}^{∗}aS_{e}, a ∈ A_{G_{A}}. Consider the C^{∗}-symbolic dynamical system (A_{G_{A}}, ρ^{A}, E_{A}). Its associated C^{∗}-algebra O_{ρ^{A}} is nothing but the Cuntz-Krieger algebra O_{A^{[2]}}. The finite directed graphs G_{A} is naturally considered to be a finite labeled graph by regarding an edge itself as its label. Hence this example will be contained in the following examples.

2. Finite labeled graphs Let G = (G, λ) be a left-resolving finite labeled graph over Σ with underlying finite directed graph G = (V, E) and labeling map λ : E → Σ. Suppose that the graph G is irreducible. Let {v_{1}, …, v_{N}} be the vertex set V. As in Section 2, we have a C^{∗}-symbolic dynamical system (A_{G}, ρ^{G}, Σ) such that A_{G} = CE_{1} ⊕ ⋯ ⊕ CE_{N} and ρ_{α}^{G}(E_{i}) = ∑_{j=1}^{N} A^{G}(i, α, j)E_{j} for i = 1, …, N, α ∈ Σ, where the N × N-matrix [A^{G}(i, α, j)]_{i,j=1,...,N} for α ∈ Σ is defined by (2.1). Put A_{G}(i, j) = ∑_{α∈Σ} A^{G}(i, α, j) for i, j = 1, …, N. Then the matrix A_{G} = [A_{G}(i, j)]_{i,j=1}^{N} is irreducible. Let r_{G} denote the Perron-Frobenius eigenvalue of the matrix A_{G}. It is easy to see that r_{G} is equal to the spectral radius r_{ρ^{G}} of the positive operator λ_{ρ^{G}}(x) = ∑_{α∈Σ} ρ_{α}^{G}(x), x ∈ A_{G}. As

λ_{ρ^{G}}(E_{i}) = ∑_{j=1}^{N} A_{G}(i, j)E_{j}, i = 1, …, N,

by identifying x = ∑_{i=1}^{N} x_{i}E_{i} ∈ A_{G} with the vector [x_{i}]_{i=1}^{N} ∈ C^{N}, one may regard the operator λ_{ρ^{G}} as the transposed matrix A_{G}^{t} of A_{G}. For a complex number β ∈ C with |β| > 1, let φ ∈ A_{G}^{∗} be a continuous linear functional belonging to E_{β}(ρ^{G}). The equality φ ∘ λ_{ρ^{G}}(E_{i}) = βφ(E_{i}) implies

∑_{j=1}^{N} A_{G}(i, j)φ(E_{j}) = βφ(E_{i}), i = 1, …, N

so that the vector [φ(E_{j})]_{j=1}^{N} is an eigenvector of A_{G} for eigenvalue β. Conversely an eigenvector [u_{i}]_{i=1}^{N} ∈ C of the matrix A_{G} for an eigenvalue β gives rise to a continuous linear functional φ on A_{G} by setting φ(E_{i}) = u_{i}, i = 1, …, N so that φ ∈ E_{β}(ρ^{G}). Hence the space E_{β}(ρ^{G}) is identified with the eigenvector space of the matrix A_{G} for eigenvalue β. Especially a faithful invariant state τ on A_{G} is the positive normalized eigenvector of A_{G} for eigenvalue r_{G}. Similarly an element x = ∑_{j=1}^{N} x_{j}E_{j} ∈ A_{G} is fixed by T_{ρ^{G}} if and only if the vector [x_{j}]_{j=1}^{N} is an eigenvector of A_{G}^{t} for the eigenvalue r_{G}. The ordinary Perron-Frobenius theorem for nonnegative matrices asserts that (A_{G}, ρ^{G}, Σ) is mean ergodic if A_{G} is irreducible. The following proposition comes from the ordinary Perron-Frobenius theorem for irreducible nonnegative matrices, which is a special case of Theorem 3.13, and Corollary 6.8.

Proposition 7.1. Suppose that the adjacency matrix A_{G} = [A_{G}(i, j)]_{i,j=1}^{N} is irreducible. Let [τ_{i}]_{i=1}^{N} and [x_{i}]_{i=1}^{N} be right and left Perron eigenvector of A_{G} respectively, that is,

A_{G}[τ_{i}]_{i=1}^{N} = r_{G}[τ_{i}]_{i=1}^{N}, A_{G}^{t}[x_{i}]_{i=1}^{N} = r_{G}[x_{i}]_{i=1}^{N},

such that ∑_{i=1}^{N} τ_{i} = 1 and ∑_{i=1}^{N} τ_{i}x_{i} = 1. Put x_{ρ^{G}} = ∑_{i=1}^{N} x_{i}E_{i} ∈ A_{G} and τ(a) = ∑_{i=1}^{N} τ_{i}a_{i} for a = ∑_{i=1}^{N} a_{i}E_{i} ∈ A_{G}. Then τ is a unique faithful invariant state on A_{G} such that the following equalities hold:

lim_{n→∞} M_{n}(a) = lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} T_{ρ^{G}}^{k}(a) = τ(a)x_{ρ^{G}}.

Furthermore the measure μ_{ρ^{G}} on D_{ρ^{G}} defined μ_{ρ^{G}}(y) = τ(yx_{ρ^{G}}) for y ∈ D_{ρ^{G}} is a unique ϕ_{ρ^{G}}-invariant measure equivalent to the measure τ on D_{ρ^{G}}.

Remark. Let X_{G} be the right one-sided sofic shift presented by G. The commutative C^{∗}-algebra C(X_{G}) on X_{G} is naturally regarded as a C^{∗}-subalgebra of D_{ρ^{G}} through the correspondence

χ_{ν} ∈ C(X_{G}) ⟶ S_{ν}S_{ν}^{∗} ∈ D_{ρ^{G}}, ν ∈ B_{k}(Λ_{G}) where χ_{ν} is the characteristic function for the cylinder

U_{ν} = {(x_{i})_{i∈N} ∈ X_{G} | x_{1} = ν_{1}, …, ν_{k} = x_{k}}.

The restriction of the ϕ_{ρ^{G}}-invariant measure μ_{ρ^{G}} on D_{ρ^{G}} to the subalgebra C(X_{G}) is nothing but a shift-invariant measure on X_{G} (cf. [21]). We will next find continuous linear functionals on O_{ρ^{G}} satisfying KMS conditions in concrete way. Now suppose that the irreducible matrix A_{G} has its period p_{G} and put

N_{G}(i, j) = {n ∈ Z_{+} | A_{G}^{n}(i, j) > 0}.

It is well-known that for n, m ∈ N_{G}(i, j) one has n ≡ m (mod p_{G}). Then for an eigenvalue β ∈ C of A_{G} with |β| = r_{G}, \frac{β}{r_{G}} is a p_{G}-th root of unity. We fix a vertex v_{1} and for k ∈ {1, 2, …, N} take n_{k} ∈ N_{G}(1, k). We set

u_{k} = (\frac{β}{r_{G}})^{n_{k}}τ(E_{k}).

Then u_{k} does not depend on the choice of n_{k} as long as n_{k} ∈ N_{G}(1, k).

Lemma 7.2. ∑_{j=1}^{N} A_{G}(i, j)u_{j} = βu_{i}, i = 1, …, N.

Proof. If A_{G}(i, j) ≠ 0, one sees n_{i} + 1 ∈ N(1, j) so that

A_{G}(i, j)u_{j} = \frac{β}{r_{G}} (\frac{β}{r_{G}})^{n_{i}}A_{G}(i, j)τ(E_{j}) = \frac{β}{r_{G}} \frac{u_{i}}{τ(E_{i})}A_{G}(i, j)τ(E_{j}).

It follows that

∑_{j=1}^{N} A_{G}(i, j)u_{j} = \frac{β}{r_{G}} \frac{u_{i}}{τ(E_{i})} ∑_{j=1}^{N} A_{G}(i, j)τ(E_{j}) = \frac{β}{r_{G}} \frac{u_{i}}{τ(E_{i})}r_{G}τ(E_{i}) = βu_{i}.

□

Hence u = [u_{k}]_{k=1}^{N} yields a nonzero eigenvector of A_{G}. Define a nonzero continuous linear functional φ on A_{G} by setting

φ(E_{k}) = u_{k}, k = 1, …, N

so that the equality φ ∘ λ_{G} = βφ on A_{G} holds. Put v_{φ} = ∑_{i=1}^{N} \frac{u_{i}}{τ(E_{i})}E_{i} ∈ A_{G}. It is easy to see that v_{φ} is a partial isometry such that φ(E_{j}) = τ(E_{j}v_{φ}), j = 1, …, N so that

φ(x) = τ(xv_{φ}), x ∈ A_{G}

holds. Therefore we have the following proposition.

Proposition 7.3. Let G = (G, λ) be a left-resolving finite labeled graph with underlying finite directed graph G = (V, E) and labeling map λ : E → Σ. Denote by {v_{1}, …, v_{N}} the vertex set V. Assume that G is irreducible. Consider the N-dimensional commutative C^{∗}-algebra A_{G} = CE_{1} ⊕⋯⊕CE_{N} where each minimal projection E_{i} corresponds to the vertex v_{i} for i = 1, …, N. Define an N ×N- nonnegative matrix A_{G} = [A_{G}(i, j)]_{i,j=1}^{N} by A_{G}(i, j) = ∑_{α∈Σ} A^{G}(i, α, j) where for α ∈ Σ and i, j = 1, …, N

A^{G}(i, α, j) = \begin{cases}1 & if there exists an edge e from v_{i} to v_{j} with λ(e) = α, \\ 0 & otherwise. \end{cases}

Let O_{A_{G}} be the associated Cuntz-Krieger algebra and τ be the unique KMS state on O_{A_{G}} for gauge action. Let β ∈ C be an eigenvalue of A_{G} such that |β| = r_{G} the Perron-Frobenius eigenvalue of the matrix A_{G}. Then a continuous linear functional on O_{A_{G}} satisfying KMS condition at Logβ is a scalar multiple of φ ∈ O_{A_{G}}^{∗} giving by for k = 1, …, N

φ(E_{k}) = (\frac{β}{r_{G}})^{n_{k}}τ(E_{k}) where n_{k} satisfies A_{G}^{n_{k}}(1, k) ≠ 0.

Consider a finite labeled graph G whose adjacency matrix A is

A = [\begin{matrix}0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 \end{matrix}] .

As

A^{3} = [\begin{matrix}2 & 2 & 0 & 0 & 0 \\ 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 2 & 4 & 0 \\ 0 & 0 & 0 & 0 & 4 \end{matrix}] ,

the period of the matrix is 3. The characteristic polynomial of A is det(t−A) = t^{2}(t^{3} − 4) so that Sp(A) = \{\sqrt[3]{4}, \sqrt[3]{4}e^{\frac{2π}{3} i}, \sqrt[3]{4}e^{\frac{4π}{3} i}, 0\} and r_{A} = \sqrt[3]{4}. Hence β ∈ Sp(A) satisfying |β| = \sqrt[3]{4} are

\sqrt[3]{4}, \sqrt[3]{4}e^{\frac{2π}{3} i}, \sqrt[3]{4}e^{\frac{4π}{3} i}.

Therefore the Cuntz-Krieger algebra O_{A} has three continuous linear functionals satisfying KMS conditions for gauge action at inverse temperatures

\frac{1}{3} log 4, \frac{1}{3} log 4 + \frac{2π}{3} i, \frac{1}{3} log 4 + \frac{4π}{3} i

respectively.

3. Dyck shifts We consider the Dyck shift D_{N} for a fixed natural number N > 1 with alphabet Σ = Σ^{−} ∪ Σ^{+} where Σ^{−} = {α_{1}, …, α_{N}}, Σ^{+} = {β_{1}, …, β_{N}}. The symbols α_{i}, β_{i} correspond to the brackets (_{i},)_{i} respectively. The Dyck inverse monoid has the relations

α_{i}β_{j} = \begin{cases}1 & if i = j, \\ 0 & otherwise \end{cases}  (7.1)

for i, j = 1, …, N (cf. [23],[26]). A word ω_{1} ⋯ ω_{n} of Σ is admissible for D_{N} precisely if ∏_{m=1}^{n} ω_{m} ≠ 0. For a word ω = ω_{1} ⋯ ω_{n} of Σ, we denote by ω̃ its reduced form. Namely ω̃ is a word of Σ ∪ {0, 1} obtained after the operations (7.1). Hence a word ω of Σ is forbidden for D_{N} if and only if ω̃ = 0. In [26], an irreducible λ-graph system presenting D_{N} called the Cantor horizon λ-graph system has been introduced. It is a minimal irreducible component of the canonical λ-graph system 𝔏^{C(D_{N})} and written as 𝔏^{Ch(D_{N})}. Let us describe the Cantor horizon λ-graph system 𝔏^{Ch(D_{N})} of D_{N}. Let Σ_{N} be the full N-shift {1, …, N}^{Z}. We denote by B_{l}(D_{N}) and B_{l}(Σ_{N}) the set of admissible words of length l of D_{N} and that of Σ_{N} respectively. The vertices V_{l} of 𝔏^{Ch(D_{N})} at level l are given by the words of length l consisting of the symbols of Σ^{+}. That is,

V_{l} = {(β_{μ_{1}} ⋯ β_{μ_{l}}) ∈ B_{l}(D_{N}) | μ_{1} ⋯ μ_{l} ∈ B_{l}(Σ_{N})}.

Hence the cardinal number of V_{l} is N^{l}. The mapping ι(= ι_{l,l+1}) : V_{l+1} → V_{l} deletes the rightmost symbol of a word in B_{l}(Σ_{N}) such as

ι((β_{μ_{1}} ⋯ β_{μ_{l+1}})) = (β_{μ_{1}} ⋯ β_{μ_{l}}), (β_{μ_{1}} ⋯ β_{μ_{l+1}}) ∈ V_{l+1}.

There exists an edge labeled α_{j} from (β_{μ_{1}} ⋯ β_{μ_{l}}) ∈ V_{l} to (β_{μ_{0}}β_{μ_{1}} ⋯ β_{μ_{l}}) ∈ V_{l+1} precisely if μ_{0} = j, and there exists an edge labeled β_{j} from (β_{j}β_{μ_{1}} ⋯ β_{μ_{l−1}}) ∈ V_{l} to (β_{μ_{1}} ⋯ β_{μ_{l+1}}) ∈ V_{l+1}. The resulting labeled Bratteli diagram with ι-map becomes a λ-graph system over Σ, denoted by 𝔏^{Ch(D_{N})}, that presents the Dyck shift D_{N} ([26]). It gives rise to a purely infinite simple C^{∗}-algebra O_{𝔏^{Ch(D_{N})}} ([32]) such that

K_{0}(O_{𝔏^{Ch(D_{N})}}) ≅ Z/NZ ⊕ C(𝔎, Z), K_{1}(O_{𝔏^{Ch(D_{N}}}) ≅ 0.

Let us denote by (A^{D_{N}} , ρ^{D_{N}} , Σ) the C^{∗}-symbolic dynamical system associated to the λ-graph system 𝔏^{Ch(D_{N})} as in Section 2. Since the vertex set V_{l} is indexed by the set B_{l}(Σ_{N}) of words, the family of projections denoted by E_{μ_{1}...μ_{l}} for μ_{1} ⋯ μ_{l} ∈ B_{l}(Σ_{N}) in the C^{∗}-algebra A^{D_{N}} forms the minimal projectins of A_{l} = C(V_{l}) such that

∑_{μ_{1}···μ_{l}∈B_{l}(Σ_{N})} E_{μ_{1}...μ_{l}} = 1, E_{μ_{1}...μ_{l}} = ∑_{μ_{l+1}=1}^{N} E_{μ_{1}...μ_{l+1}}.

As the algebra A_{l} is embedded into A_{l+1}, the C^{∗}-algebra A^{D_{N}} is a commutative AF-algebra generated by the subalgebras A_{l}, l ∈ N. The endomorphisms ρ_{γ}^{D_{N}} : A^{D_{N}} ⟶ A^{D_{N}} for γ ∈ Σ are defined by

ρ_{α_{j}}^{D_{N}} (E_{μ_{1}...μ_{l}}) = E_{jμ_{1}...μ_{l}}, ρ_{β_{j}}^{D_{N}} (E_{jμ_{1}...μ_{l−1}}) = ∑_{μ_{l},μ_{l+1}=1}^{N} E_{μ_{1}...μ_{l+1}}

for μ_{1} … μ_{l} ∈ B_{l}(Σ_{N}) and j = 1, …, N. It then follows that

λ_{ρ^{D_{N}}} (1) = ∑_{j=1}^{N} ρ_{α_{j}}^{D_{N}} (1) + ∑_{j=1}^{N} ρ_{β_{j}}^{D_{N}} (1)

= ∑_{j=1}^{N} ∑_{μ_{1}···μ_{l}∈B_{l}(Σ_{N})} E_{jμ_{1}...μ_{l}} + ∑_{j=1}^{N} ∑_{μ_{2}···μ_{l}∈B_{l−1}(Σ_{N})} ∑_{μ_{l+1},μ_{l+2}=1}^{N} E_{μ_{2}...μ_{l+2}}
= 1 + N

so that we have ∥λ_{ρ^{D_{N}}} ∥ = ∥λ_{ρ^{D_{N}}} (1)∥ = 1 + N. Hence we obtain

r_{ρ^{D_{N}}} = 1 + N, T_{ρ^{D_{N}}} (1) = 1.

This implies that 1 is a fixed element by T_{ρ^{D_{N}}} and hence (A^{D_{N}} , ρ^{D_{N}} , Σ) satisfies (FP). As in [32], (A^{D_{N}} , ρ^{D_{N}} , Σ) is irreducible and uniquely ergodic, so that it is mean ergodic. One then sees that there exists a KMS state at inverse temperature log β if and only if β = 1 + N. The admitted KMS state is unique ([32, Theorem 1.2]).

4. β-shifts Let β > 1 be an arbitrary real number. Take a natural number N with N −1 < β ≤ N. Put Σ = {0, 1, ..., N − 1}. For a nonnegative real number t, we denote by [t] the integer part of t. Let f_{β} : [0, 1] → [0, 1] be the mapping defined by

f_{β}(x) = βx − [βx], x ∈ [0, 1]

that is called the β-transformation ([38], [42]). The β-expansion of x ∈ [0, 1] is a sequence {d_{i}(x, β)}_{i∈N} of integers of Σ determined by

d_{i}(x, β) = [βf_{β}^{i−1}(x)], i ∈ N.

By this sequence, we can write x as

x = ∑_{i=1}^{∞} \frac{d_{i}(x, β)}{β^{i}}.

We endow the infinite product Σ^{N} with the product topology and the lexicographical order. Put ζ_{β} = sup_{x∈[0,1)}(d_{i}(x, β))_{i∈Σ^{N}}. We define the shiftinvariant compact subset X_{β} of Σ^{N} by

X_{β} = {ω ∈ Σ^{N}|σ^{i}(ω) ≤ ζ_{β}, i = 0, 1, 2, ...},

where σ denotes the shift σ((ω_{i})_{i∈N}) = (ω_{i+1})_{i∈N}. The one-sided subshift (X_{β}, σ) is called the right one-sided β-shift (cf. [38], [42]). Its (two-sided) subshift

Λ_{β} = {(ω_{i})_{i∈Z} ∈ Σ^{Z} | (ω_{i−k})_{i∈N} ∈ X_{β}, k = 0, 1, 2, … }

is called the β-shift. In [17], the C^{∗}-algbera O_{β} associated with the β-shift has been introduced and studied. It is simple and purely infinite for every β > 1 and generated by N − 1 isometries S_{0}, S_{1}, …, S_{N−2} and one partial isometry S_{N−1} having certain operator relations (see [17]). The family O_{β}, 1 < β ∈ R interpolates the Cuntz algebras O_{n}, 1 < n ∈ N. Denote by A_{β} the C^{∗}- subalgebra of O_{β} genertaed by the family of the projections S_{μ}^{∗}S_{μ}, μ ∈ B_{∗}(Λ_{β}). The algebra is commutative and is of infinite dimensional unless Λ_{β} is sofic, where Λ_{β} is sofic if and only if the sequence (d_{i}(1, β))_{i∈N} is ultimately periodic. Define a family {ρ_{j}^{β}}_{j=0,1,...,N−1} of endomorphisms on A_{β} by

ρ_{j}^{β}(x) = S_{j}^{∗}xS_{j}, x ∈ A_{β}, j = 0, 1, …, N − 1

so that we have a C^{∗}-symbolic dynamical system (A_{β}, ρ^{β}, Σ). It is direct to see that the C^{∗}-algebra O_{ρ^{β}} is canonically isomorphic to the C^{∗}-algebra O_{β}. We set the positive operator λ_{β} on A_{β} by

λ_{β}(x) = ∑_{j=0}^{N−1} ρ_{j}^{β}(x), x ∈ A_{β}.

Lemma 7.4. The spectral radius r_{β} of the positive operator λ_{β} on A_{β} is β.

Proof. Denote by θ_{k} the cardinal number of the admissible words B_{k}(Λ_{β}) of length k. Then we have

∥λ_{β}^{k}∥ = ∥λ_{β}^{k}(1)∥ ≤ ∑_{μ∈B_{k}(Λ_{β})} ∥S_{μ}^{∗}S_{μ}∥ = θ_{k}.

As in [44, p. 179], lim_{k→∞} \frac{θ_{k}}{β^{k}} converges to a positive real number so that there

exists a positive constant M > 0 such that \frac{∥λ_{β}^{k}∥}{β^{k}} < M for all k ∈ N. Hence lim_{k→∞} ∥λ_{β}^{k}∥^{\frac{1}{k}} ≤ β so that r_{β} ≤ β. As in [17], there exists a state τ on A_{β} satisfying τ ∘ λ_{β} = βτ. This implies β ∈ Sp(λ_{β}) so that r_{β} = β. □

Proposition 7.5. (A_{β}, ρ^{β}, Σ) is irreducible, uniquely ergodic and powerbounded.

Proof. It has been proved in [17] that there is no nontrivial ideal of A_{β} invariant under λ_{β} and there exists a unique state τ on A_{β} satisfying τ ∘λ_{β} = r_{β}τ. Hence (A_{β}, ρ^{β}, Σ) is irreducible, uniquely ergodic. As in the proof of the above lemma,

there exists a positive constant M > 0 such that \frac{∥λ_{β}^{k}∥}{r_{β}^{k}} < M for all k ∈ N. This

means that (A_{β}, ρ^{β}, Σ) is power-bounded. □

By the above proposition, one knows that (A_{β}, ρ^{β}, Σ) satisfies the hypothesis of Theorem 6.7 so that there uniquely exists a ϕ_{ρ^{β}}-invariant measure on D_{ρ^{β}} absolutely continuous with respect to the restriction of the unique KMS-state τ to D_{ρ^{β}}. We note that C(X_{β}) is a C^{∗}-subalgebra of D_{ρ^{β}} and the restriction of ϕ_{ρ^{β}} to C(X_{β}) comes from the shift transformation σ. As in [17], the restriction of the KMS-state τ to D_{ρ^{β}} corresponds to the Lebesgue measure on [0, 1] in translating the β-shift to the β-transformation. Hence the uniqueness of the ϕ_{ρ^{β}}-invariant measure on D_{ρ^{β}} absolutely continuous with respect to τ exactly corresponds to the uniqueness of the invariant measure on [0, 1] under the β-transformation absolutely continuous with respect to the Lebesgue measure studied in [14], [38] and [42]. In fact, the density function h_{β} appeared in [14], [38] and [42] of the invariant measure for the β-transformation with respect to the Lebesgue measure is the element x_{ρ^{β}} realized as the mean lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} \frac{λ_{β}^{k}(1)}{β^{k}} in Theorem 6.7.

Acknowledgments: The author would like to thank Yasuo Watatani who pointed out inaccuracy in the proof of Lemma 3.4 in the first draft of the paper. The author is also deeply indebted to the referee whose many useful suggestions, guidences and comments improved the paper. This work was supported by JSPS Grant-in-Aid for Scientific Reserch ((C), No 20540215).

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Kengo Matsumoto

Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan kengo@juen.ac.jp 176

Documenta Mathematica 16 (2011)