Lobachevskii Journal of Mathematics

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Vol. 14, 2004, 17–24 \begin{matrix}c \\ ◯ \end{matrix} A.M. Bikchentaev

A.M. Bikchentaev

THE CONTINUITY OF MULTIPLICATION FOR TWO TOPOLOGIES ASSOCIATED WITH A SEMIFINITE TRACE ON VON NEUMANN ALGEBRA

(submitted by D. Mushtari)

Abstract. Let M be a semifinite von Neumann algebra in a Hilbert space H and τ be a normal faithful semifinite trace on M. Let M^{pr} denote the set of all projections in M, e denote the unit of M, and ∥ · ∥ denote the C^{∗}-norm on M.

The set of all τ-measurable operators M̃ with sum and product defined as the respective closures of the usual sum and product, is a *-algebra. The sets

U(ε, δ)={x ∈ M̃: ∥xp∥ ≤ ε and τ(e − p) ≤ δ for some p ∈ M^{pr}}, ε>0, δ >0,

form a base at 0 for a metrizable vector topology t_{τ} on M̃, called the measure topology. Equipped with this topology, M̃ is a complete topological *-algebra. We will write x_{i} ⟶^{τ} x in case a net {x_{i}}_{i∈I} ⊂ M̃ converges to x ∈ M̃ for the measure topology on M̃. By definition, a net {x_{i}}_{i∈I} ⊂ M̃ converges τ-locally to x ∈ M̃ (notation: x_{i} ⟶^{τl} x) if x_{i}p ⟶^{τ} xp for all p ∈ M^{pr}, τ(p) < ∞; and a net {x_{i}}_{i∈I} ⊂ M̃ converges weak τ-locally to x ∈ M̃ (notation: x_{i} ⟶^{wτl} x) if

px_{i}p ⟶^{τ} pxp for all p ∈ M^{pr}, τ(p) < ∞.

Theorem 1. Let x_{i}, x ∈ M̃.

1. If x_{i} ⟶^{τl} x, then x_{i}y ⟶^{τl} xy and yx_{i} ⟶^{τl} yx for every fixed y ∈ M̃.

2000 Mathematical Subject Classification. 46L51, 46L53, 46E30, 46H05. Key words and phrases. Hilbert space, von Neumann algebra, noncom-

mutative integration, measurable operator, semifinite trace, convergence with respect to measure, compact operator, topological algebra. Partially supported by Russian Foundation for Basic Research (Grant 01-

01-00129) and the scientific program ”Universities of Russia – Basic Research” (Grant UR. 04.01.061) .

2. If x_{i} ⟶^{wτl} x, then x_{i}y ⟶^{wτl} xy and yx_{i} ⟶^{wτl} yx for every fixed y ∈ M̃.

Theorem 2. If {x_{i}}_{i∈I} ⊂ M̃ is bounded in measure and if x_{i} ⟶^{τl} x ∈ M̃, then x_{i}y ⟶^{τ} xy for all τ-compact y ∈ M̃.

Theorem 3. Let x, y, x_{i}, y_{i} ∈ M̃ and let a set {x_{i}}_{i∈I} be bounded in mea-

sure. If x_{i} ⟶^{τl} x and y_{i} ⟶^{τl} y, then x_{i}y_{i} ⟶^{τl} xy.

If M is abelian, then the weak τ-local and τ-local convergencies on M̃ coincides with the familiar convergence locally in measure. If τ(e) = ∞, then the boundedness condition cannot be omitted in Theorem 2.

If M is B(H) with standard trace, then Theorem 2 for sequences is a ”Basic lemma”of the theory of projection methods: If y is compact and x_{n} → x strongly, then x_{n}y → xy uniformly, i.e. ∥x_{n}y−xy∥ → 0 as n → ∞. Theorem 3 means that the mapping

(x, y) ↦ xy : (B(H)_{1} × B(H) → B(H))

is strong-operator continuous (B(H)_{1} denotes the unit ball of B(H)).

The author is greatly indebted to O. E. Tikhonov for drawing author’s attention to the problem of the τ-local continuity of operator functions.

1. Introduction

Let M be a semifinite von Neumann algebra of operators in a Hilbert space H and τ be a distinguished normal faithful semifinite trace on M. Let M^{pr} denote the lattice of all projections in M, e denote the identity, and M_{1} denote the unit ball of M in the C^{∗}-norm ∥ · ∥ on M. The closed, densely defined linear operator x in H with domain D(x) is said to be affiliated with M if and only if u^{∗}xu = x for all unitary operators u in the commutant M′ of M. If x is affiliated with M then x is said to be τ-measurable if and only if, for every ε > 0 there exists a projection p ∈ M^{pr} for which p(H) ⊆ D(x) and τ(e − p) < ε. We denote by M̃ the set of all τ-measurable operators. With sum and product defined as the respective closures of the usual sum and product, M̃ is a *-algebra. The sets

U(ε, δ)={x∈M̃: ∥xp∥ ≤ ε and τ(e − p) ≤ δ for some p ∈ M^{pr}},

where ε > 0, δ > 0, form a base at 0 for a metrizable vector topology t_{τ} on M̃, called the measure topology ([8]; [11, p. 18]). Equipped with this topology, M̃ is a complete topological *-algebra in which M is dense. We will write x_{i} ⟶^{τ} x in case a net {x_{i}}_{i∈I} ⊂ M̃ converges to x ∈ M̃ for the measure topology on M̃.

A subset X of M̃ is bounded in measure, if it is bounded with respect to this topology on the vector space of M̃, that is in case for every neighborhood U of 0 there is an α > 0 such that αX ⊂ U [8, p. 106].

If M is B(H), the von Neumann algebra of all bounded linear operators in H is equipped with the usual standard trace, then M̃ coincides with M and in this case the measure topology coincides with the ∥·∥-topology. If M is abelian, then M may be identified with L^{∞}(Ω, μ) and τ(f) = ∫_{Ω} f dμ where (Ω, μ) is a localizable measure space. In this case, M̃ is the space S_{0}(Ω) consisting of those measurable complex-valued functions on Ω which are bounded except on a set of finite measure and the measure topology on M̃ may be identified simply with the familiar topology of convergence in measure.

If x is any self-adjoint operator in H and if

x = ∫_{ℝ} λ de_{λ}^{x}

is its spectral representation, we will write χ_{T} (x) for the spectral projection of x corresponding to the Borel subset T ⊂ ℝ. In particular e_{λ}^{x} = χ_{(−∞,λ]}(x). If x is closed, densely defined linear operator affiliated with M and |x| = \sqrt{x^{∗}x}, then the spectral resolution χ_{•}(|x|) is contained in M and x ∈ M̃ if and only if there exists λ ∈ ℝ such that τ(χ_{(λ,∞)}(|x|)) < ∞.

For p, q ∈ M^{pr} we write p ∼ q (the Murray – von Neumann equivalence), if u^{∗}u = p and uu^{∗} = q for some u ∈ M.

A linear set D in H is said to be associated with M if u(D) ⊂ D for every unitary operator u in M′. If D is a closed linear manifold then D is associated with M if and only if the projection onto D lies in M [9, p. 403]. For every x ∈ M̃ the projection onto the closure of the range of x lies in M. It is equal to the left support projection

s_{l}(x) = ∧{q ∈ M^{pr} : qx = x}

and s_{l}(x) ∼ s_{l}(x^{∗}).

The two-sided ideal of τ-compact operators

M̃_{0} = {x ∈ M̃: τ(χ_{(λ,∞)}(|x|)) < ∞ for all λ > 0}

is closed in measure topology [12]. If M is B(H) with standard trace, then M̃_{0} is precisely the ideal of compact operators. Let

M_{0}^{pr} = M̃_{0} ⋂ M^{pr} = {p ∈ M^{pr} : τ(p) < ∞}.

Definition 1 (cf. [3, p. 114]). A net {x_{i}}_{i∈I} ⊂ M̃ is said to converge τ-locally to x ∈ M̃ (notation: x_{i} ⟶^{τl} x) if x_{i}p ⟶^{τ} xp for all p ∈ M_{0}^{pr}.

Definition 2 (cf. [3, p. 114]; [5, p. 746]). A net {x_{i}}_{i∈I} ⊂ M̃ is said to converge weak τ-locally to x ∈ M̃ (notation: x_{i} ⟶^{wτl} x) if px_{i}p ⟶^{τ} pxp for all p ∈ M_{0}^{pr}.

It is clear that

x_{i} ⟶^{τ} x ⟹ x_{i} ⟶^{τl} x ⟹ x_{i} ⟶^{wτl} x for x_{i}, x ∈ M̃.

If M is B(H) with standard trace, then τ-local (respectively, weak τ-local) convergence coincides with strong-operator (respectively, weakoperator) convergence. If τ(e) < ∞, then M̃ consists of all densely defined closed linear operators affiliated with M and weak τ-local convergence is precisely the convergence in measure topology on M̃. Moreover, the measure topology is a minimal one in the class of all topologies which are Hausdorff, metrizable, and compatible with the ring structure of M̃ [1, Theorem 2].

2. Main Results

Further we assume that τ(e) = ∞.

Theorem 1. Let x_{i}, x ∈ M̃.

1. If x_{i} ⟶^{τl} x, then x_{i}y ⟶^{τl} xy and yx_{i} ⟶^{τl} yx for every fixed y ∈ M̃.

2. If x_{i} ⟶^{wτl} x, then x_{i}y ⟶^{wτl} xy and yx_{i} ⟶^{wτl} yx for every fixed y ∈ M̃.

Proof. Let x_{i}, x, y ∈ M̃ and let p ∈ M_{0}^{pr}. Since s_{l}(yp) ∼ s_{l}(py^{∗}) ≤ p, one has s_{l}(yp) ∈ M_{0}^{pr}.

1. Suppose that x_{i} ⟶^{τl} x. One has

yx_{i} ⟶^{τl} yx and x_{i}yp = x_{i}s_{l}(yp)yp ⟶^{τ} xs_{l}(yp)yp = xyp,

since the multiplication operations z ↦ yz (M̃ → M̃) and z ↦ zyp (M̃ → M̃) are continuous in the measure topology.

2. One has r = p ∨ q ∈ M_{0}^{pr} for p, q ∈ M_{0}^{pr}, since p ∨ q − p ∼ q − p ∧ q [8, p. 105]. By [3, p. 114] x_{i} ⟶^{wτl} x if and only if px_{i}q ⟶^{τ} pxq for all p, q ∈ M_{0}^{pr}. Indeed, from rx_{i}r ⟶^{τ} rxr it follows that

px_{i}q = p · rx_{i}r · q ⟶^{τ} p · rxr · q = pxq.

Therefore,

px_{i}yp = px_{i}s_{l}(yp)yp ⟶^{τ} pxs_{l}(yp)yp = pxyp.

Now the convergence yx_{i} ⟶^{wτl} yx follows from the fact that the mapping z ↦ z^{∗} (M̃ → M̃) is weak τ-local continuous and by taking adjoints.

Theorem 2. If {x_{i}}_{i∈I} ⊂ M̃ is bounded in measure and if x_{i} ⟶^{τl} x ∈

M̃, then x_{i}y ⟶^{τ} xy for all y ∈ M̃_{0}.

Proof. Step 1. Without loss of generality we may assume that y ∈ M̃_{0} is self-adjoint and non-negative. Indeed, let y ∈ M̃_{0} and y^{∗} = u|y^{∗}| be the polar decomposition of y^{∗}. Then y = |y^{∗}|u^{∗} and from x_{i}|y^{∗}| ⟶^{τ} x|y^{∗}| it follows that x_{i}y ⟶^{τ} xy, since the multiplication operation z ↦ zu^{∗} (M̃ → M̃) is continuous in the measure topology.

Step 2. Fix non-negative y ∈ M̃_{0} and ε, δ > 0. A subset X of M̃ is bounded in measure if and only if for every d > 0 there exists a constant c < ∞ such that X ⊂ U(c, d) [8, p. 106]. Let n ∈ ℕ and

y_{1,n} = ∫_{[0,n^{−1})} λ de_{λ}^{y}, y_{2,n} = ∫_{[n^{−1},n)} λ de_{λ}^{y}, y_{3,n} = ∫_{[n,∞)} λ de_{λ}^{y}.

Then y = y_{1,n} + y_{2,n} + y_{3,n} and for z_{i} = x_{i} − x one has

x_{i}y − xy = z_{i}y_{1,n} + z_{i}y_{2,n} + z_{i}y_{3,n}, i ∈ I.  (1)

The set {z_{i}}_{i∈I} is bounded in measure. There exists a constant c > 0 such that

{z_{i}}_{i∈I} ⊂ U(c, δ).  (2)

Let

n_{1} = min{k ∈ ℕ : 2εk ≥ c}.

Since ∥y_{1,n}∥ < n^{−1}, one has y_{1,n} ⟶^{τ} 0 as n → ∞. Since

τ(s_{l}(y_{3,n})) = τ(χ_{[n,∞)}(y)) → 0 as n → ∞,

one has y_{3,n} ⟶^{τ} 0 as n → ∞. Therefore y_{1,n} + y_{3,n} ⟶^{τ} 0 as n → ∞. Then there exists m ∈ ℕ such that

y_{1,n} + y_{3,n} ⊂ U(n_{1}^{−1}, δ) for all n ≥ m.  (3)

Recall that

U(ε_{1}, δ_{1})U(ε_{2}, δ_{2}) ⊂ U(ε_{1}ε_{2}, δ_{1} + δ_{2}) for all ε_{1}, δ_{1}, ε_{2}, δ_{2} > 0 (4)

by [8, p. 107], [11, p. 18]. Now by (2) and (3) one has

z_{i}y_{1,n} + z_{i}y_{3,n} ∈ U(2ε, 2δ) for all i ∈ I, n ≥ m.  (5)

Step 3. Let m be as above, λ_{k} > 0 and p_{k} ∈ M_{0}^{pr} (k = 1, …, j), p_{k}p_{l} = 0 for k ≠ l, such that

y_{2,m}^{2} = ∫_{[m^{−1},m)} λ^{2} de_{λ}^{y} ≤ ∑_{k=1}^{j} λ_{k}^{2}p_{k}

(one can choose p_{k} as spectral projections of y). There exists z ∈ M_{1} such that

y_{2,m} = (∑_{k=1}^{j} λ_{k}p_{k}) · z

[4, Chap. 1, Sect. 1, Lemma 2]. Since z_{i} ⟶^{τl} 0, one has z_{i}p_{k} ⟶^{τ} 0 for all k = 1, …, j. Now

z_{i}y_{2,m} = ∑_{k=1}^{j} λ_{k}z_{i}p_{k}z ⟶^{τ} 0,

because the multiplication operation t ↦ tz (M̃ → M̃) is continuous in the measure topology. Therefore, there exists i_{0} ∈ I such that

z_{i}y_{2,m} ∈ U(ε, δ) for all i ∈ I, i ≥ i_{0}.  (6)

Step 4. Recall that

U(ε_{1}, δ_{1}) + U(ε_{2}, δ_{2}) ⊂ U(ε_{1} + ε_{2}, δ_{1} + δ_{2}) for all ε_{1}, δ_{1}, ε_{2}, δ_{2} > 0 (7)

by [8, p. 107], [11, p. 18]. The assertion of Theorem 2 follows from (1),

(5) and (6), since

x_{i}y − xy ∈ U(3ε, 3δ) for all i ∈ I, i ≥ i_{0}.

Theorem 3. Let x, y, x_{i}, y_{i} ∈ M̃ and let a set {x_{i}}_{i∈I} be bounded in

measure. If x_{i} ⟶^{τl} x and y_{i} ⟶^{τl} y, then x_{i}y_{i} ⟶^{τl} xy.

Proof. For every p ∈ M_{0}^{pr} one has

x_{i}y_{i}p − xyp = x_{i}(y_{i}p − yp) + (x_{i} − x)yp, i ∈ I.  (8)

Fix ε, δ > 0. By assumption of the theorem, there exists a constant c > 0 such that

{x_{i}}_{i∈I} ⊂ U(c, δ).  (9)

Since y_{i}p − yp ⟶^{τ} 0, there exists i_{1} ∈ I such that

y_{i}p − yp ∈ U(2εc^{−1}, δ) for all i ∈ I, i ≥ i_{1}.  (10)

Now by (9), (10) and (4) one has

x_{i}(y_{i}p − yp) ∈ U(2ε, 2δ) for all i ∈ I, i ≥ i_{1}.  (11)

Since x_{i} − x ⟶^{τl} 0 and yp ∈ M̃_{0} , it follows by Theorem 2 that there exists i_{2} ∈ I such that

(x_{i} − x)yp ∈ U(ε, δ) for all i ∈ I, i ≥ i_{2}.  (12)

There exists i_{0} ∈ I such that i_{0} ≥ i_{1} and i_{0} ≥ i_{2}. Now by (8), (11), (12) and (7) one has

x_{i}y_{i}p − xyp ∈ U(3ε, 3δ) for all i ∈ I, i ≥ i_{0}.

This proves the theorem.

Example 1. If M is abelian, then the weak τ-local and τ-local convergencies on M̃ coincides with the familiar convergence locally in measure (i.e., in other words, convergence in measure on every set of finite measure). The boundedness condition for {x_{i}}_{i∈I} cannot be omitted in Theorem 2. Indeed, let Ω = (0, ∞) equipped with the Lebesgue measure μ. Define the functions

y(t) = min{1, t^{−1}}; x_{n}(t) = t χ_{[n,2n]}(t) (t ∈ (0, ∞), n ∈ ℕ).

Then

i) x_{n} ⟶^{τl} 0 as n → ∞;

ii) {x_{n}}_{n=1}^{∞} is not bounded in measure;

iii) y ∈ M̃_{0} ⋂ M_{1};

iv) since (x_{n}y)(t) = χ_{[n,2n]}(t) for every t ∈ (0, ∞), n ∈ ℕ, x_{n}y does not converge in measure topology.

Example 2. If M is B(H) with standard trace, then Theorem 2 for sequences is a ”Basic lemma”of the projection methods [2, pp. 18–19] (the boundedness condition for {x_{n}}_{n=1}^{∞} follows from the principle of uniform boundedness):

If y is compact and x_{n} → x strongly, then x_{n}y → xy uniformly, i.e. ∥x_{n}y − xy∥ → 0 as n → ∞.

Theorem 3 means that the mapping

(x, y) ↦ xy : (B(H)_{1} × B(H) → B(H))

is strong-operator continuous [7, pp. 115–117].

Remark. The second part of Theorem 1 was already used in [6] and 24 A.M. BIKCHENTAEV

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Research Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya str. 17, Kazan:420008, RUSSIA E-mail address: Airat.Bikchentaev@ksu.ru

Received January 8, 2004