Lobachevskii Journal of Mathematics Vol. 6, 2000, 33–37 \begin{matrix}c \\ ◯ \end{matrix}B. T. Batikyan

B. T. Batikyan

POINT DERIVATIONS ON ALGEBRAIC EXTENSION OF BANACH ALGEBRA

(submitted by M.A.Malakhaltsev)

Abstract. It is shown that a point y_{0} in the carrier space of algebraic extension B of commutative Banach algebra A is a branch point if and only if there exists a local point derivation on B at y_{0}, whose kernel contains A.

Throughout this paper A will denote a unital commutative Banach algebra with a carrier space X and A[t] will denote the algebra of polynomials in one indeterminate t over A. To every polynomial

q = a_{0}t^{m} + a_{1}t^{m−1} + ⋯ + a_{m}  (1)

in A[t] we assign a continuous function

q̂(x, λ) = ∑_{k=0}^{m} â_{k}(x)λ^{m−k},  (2)

on X × ℂ and a polynomial with complex coefficients

q_{x} = ∑_{k=0}^{m} â_{k}(x)t^{m−k},  (3)

where â_{k} is the Gelfand transform of element a_{k} ∈ A, 0 ≤ k ≤ m, and x is an arbitrary point of X.

The set

S(q) = {(x, λ) ∈ X × ℂ : q_{x} ≠ 0, q̂(x, λ) = 0}  (4)

is called the spectrum of polynomial q. If a point y_{0} = (x_{0}, λ_{0}) belongs to S(q) then the multiplicity of λ_{0} as a root of equation q_{x_{0}} = 0 is denoted by μ_{0}. We call μ_{0} the multiplicity of y_{0}.

Let us consider the continuous surjective mapping

π_{q} : S(q) → X, π_{q}(x, λ) = x.  (5)

By Lindberg’s theorem (see [1]), there exists a base {W } for S(q) at each point y_{0} = (x_{0}, λ_{0}) ∈ S(q) such that the sum of multiplicities of all points in π_{q}^{−1}(x) ∩ W is equal to μ_{0} for every x ∈ π_{q}(W). In particular, the map π_{q} is a finite-sheeted covering map, and if μ_{0} = 1, then π_{q} is a local homeomorphism at y_{0}.

Let us fix a monic polynomial

p = t^{n} + c_{1}t^{n−1} + ⋯ + c_{n}  (6)

of degree n > 1 with coefficients in A. We can assume without loss of generality that

∑_{k=1}^{n} ||c_{k}|| ≤ 1.  (7)

Let 𝔭 be a principal ideal of A[t] generated by polynomial p. A quotient algebra B = A[t]/𝔭 is called an algebraic extension of A generated by p. The algebra B consists of all polynomials over A, whose degrees are at most n − 1, and the multiplication in B is determined by the following identity:

t^{n} = −c_{1}t^{n−1} − ⋯ − c_{n}.  (8)

It is well known that B is a commutative Banach algebra with respect to the norm given by

∥∑_{k=0}^{n−1} a_{k}t^{k}∥ = ∑_{k=0}^{n−1} ||a_{k}||,  (9)

Moreover, the canonical epimorphism ρ : A[t] → B induces the isometric embedding of A into B. The carrier space Y of algebra B is identified with a compact space S(p), and π_{p} is a covering mapping of Y onto X. The Gelfand transform of an element b ∈ B coincides with the restriction of function b̂(x, λ) to S(p). (For a detailed discussion of properties of an algebraic extension of a commutative Banach algebra, the reader is referred to [2] and [3], for example.)

A point y_{0} in the carrier space of algebra B is said to be critical if its multiplicity μ_{0} is greater than 1, or what is equivalent, y_{0} belongs to S(p′), where p′ is the formal derivative of polynomial p with respect to t. If a point y_{0} does not have a neighborhood V such that the restriction of π_{p} to V is injective then y_{0} is called a branch point. Note that the set of all branch points is closed, nowhere dense in S(p) and, as we have pointed above, it is contained (in general, strictly) in the set of all critical points.

Let d be a point derivation on an algebra B at a point y_{0} = (x_{0}, λ_{0}), that is, d is a linear functional on A such that

d(b_{1}b_{2}) = d(b_{1})b̂_{2}(x_{0}, λ_{0}) + d(b_{2})b̂_{1}(x_{0}, λ_{0})  (10)

for all b_{1}, b_{2} in B. If d(b) = 0 for each b ∈ B, such that a function b̂ vanishes in a neighborhood of y_{0} in S(p) then a derivation d is said to be local. It should be noted that there exist point derivations which are not local (The general method of construction of non-local derivations on a uniform algebra can be found in [4]).

We shall denote by Ω(B, y_{0}) and Ω(A, x_{0}) spaces of all point derivations on B at a point y_{0} and on A at a point x_{0} respectively. Let Ω_{A}(B, y_{0}) be a subspace of Ω(B, y_{0}) consisting of all point derivations whose kernels contain

A. If D ∈ Ω_{A}(B, y_{0}) and b = ∑_{k=0}^{n−1} a_{k}t^{k}, then

D(b) = ∑_{k=1}^{n−1} â_{k}(x_{0})kD(t)λ_{0}^{k−1} = D(t)b̂′(x_{0}, λ_{0}).  (11)

Since every functional in Ω_{A}(B, y_{0}) is a morphism between A-modules B and ℂ, we shall call it an A-derivation. It follows from (11) that each A-derivation D is automatically continuous and ||D|| = (n − 1)|D(t)|. It is easily seen that for every point y_{0} ∈ S(p) a subspace Ω_{A}(B, y_{0}) is either trivial or one-dimensional.

Theorem 1. A point y_{0} ∈ S(p) is critical if and only if a subspace Ω_{A}(B, y_{0}) is nontrivial.

Proof. If y_{0} = (x_{0}, λ_{0}) is a critical point, then r̂′(x_{0}, λ_{0}) = 0 for each polynomial r ∈ 𝔭. This means that the functional q → q̂′(x_{0}, λ_{0}), defined on A[t] induces a nontrivial A-derivation on B at the point y_{0}. Thus Ω_{A}(B, y_{0}) is nontrivial.

Conversely, let D ∈ Ω_{A}(B, y_{0}) and D(t) = 1. Then D̃ = D ∘ ρ is an Aderivation on A[t] at the point y_{0} and hence D̃(q) = q̂′(x_{0}, λ_{0}) for each q ∈ A[t]. In particular, p̂′(x_{0}, λ_{0}) = D̃(p) = D(0) = 0. This completes the proof of the theorem.

Now we will show that A-derivation characterize branch points as well.

Theorem 2. Let y_{0} be a critical point and D (D(t) = 1) be a point Aderivation on B at y_{0}. Then the point y_{0} is a branch point in the carrier space of B if and only if the derivation D is local.

Proof. Suppose that y_{0} = (x_{0}, λ_{0}) is a branch point and a polynomial b =

∑_{k=0}^{n−1} a_{k}t^{k} is such that the function b̂ vanishes in a neighborhood V ⊂ S(p)

of the point y_{0}. Obviously, if all functions â_{k} vanish at the point x_{0}, then D(b) = 0. On other hand, if â_{k}(x_{0}) ≠ 0 for some k > 0, then there exists a neighborhood W ⊂ V of the point y_{0}, which is contained in S(b) and such that π_{b}|W = π_{p}|W. By Lindberg’s theorem, if D(b) = b̂′(x_{0}, λ_{0}) ≠ 0, then π_{b} is a local homeomorphism at y_{0}. Hence, π_{p} is also a local homeomorphism at y_{0}, but this is impossible.

Conversely, to obtain a contradiction, we suppose that y_{0} is not a branch point and μ_{0} > 1 is a multiplicity of y_{0}. Then there exists a neighborhood V ⊂ S(p) of the point y_{0} such that the multiplicity of every point in V is equal to μ_{0}. Let b be the (μ_{0} − 1)th formal derivative of the polynomial p with respect to t. It is clear that b ∈ B and the function b̂ vanishes in V, but D(b) ≠ 0. This contradiction completes the proof of the theorem.

Since algebra A is embedded into algebra B, it is natural to study the problem of extending a point derivation on A to a derivation on B (Cf. the problem of extending a derivation operator in the theory of field extensions [5 Ch. X, §7]).

Let d ∈ Ω(A, x_{0}) and let y_{0} = (x_{0}, λ_{0}) belong to π_{p}^{−1}(x_{0}). It is easily seen that any extension of d to a point derivation on A[t] at y_{0} is given by the formula:

d_{β}(q) = ∑_{k=0}^{m} d(a_{k})λ_{0}^{m−k} + βq̂′(x_{0}, λ_{0}),  (12)

where q = ∑_{k=0}^{m} a_{k}t^{m−k} and β ∈ ℂ. A functional d_{β} is a point derivation on B if

and only if d_{β}(p) = 0, i.e.

βp̂′(x_{0}, λ_{0}) = −d(c),  (13)

where

c = λ_{0}^{n} + c_{1}λ_{0}^{n−1} + ⋯ + c_{n}  (14)

is a fixed element of the algebra A.

In particular, if y_{0} is not a critical point, then

β = −\frac{d(c)}{p̂′(x_{0}, λ_{0})},  (15)

and there is a unique extension of d to a derivation on B.

Now let p̂′(x_{0}, λ_{0}) = 0. We denote by Ω_{0}(A, x_{0}) a subspace of Ω(A, x_{0}) consisting of derivations which vanish at c. If d ∉ Ω_{0}(A, x_{0}), then there is no a derivation extending d on B. If d(c) = 0, then any d_{β} is an extension of d to a derivation on B, moreover, if β ≠ γ, then d_{β} − d_{γ} is a point A-derivation.

We summarize these results in the following theorem.

Theorem 3. Let B be an algebraic extension of an algebra A, let x_{0} ∈ X and y_{0} ∈ π_{p}^{−1}(x_{0}). Then

1. If y_{0} is not a critical point, then the spaces Ω(A, x_{0}) and Ω(B, y_{0}) are isomorphic;

2. If y_{0} is a critical point, then Ω(B, y_{0}) = Ω_{0}(A, x_{0}) ⊕ Ω_{A}(B, y_{0}).

In particular, dimensions of spaces Ω(B, y_{0}) and Ω(A, x_{0}) either coincide or dim Ω(B, y_{0}) = dim Ω(A, x_{0}) + 1. It should be noted that if a point derivation d is continuous on A, then any its extension to a derivation on B is also continuous.

References

[1] Lindberg J.A., ”Factorization of polynomials over Banach algebras.” Trans. Amer. Math. Soc. 1964. V. 112. P. 356–368.

[2] Arens R., Hoffman K., ”Algebraic extensions of normed algebras.” Proc. Amer. Math. Soc. 1956. V. 7. P. 203–210.

[3] Lindberg J.A., ”Algebraic extensions of commutative Banach algebras.” Pacif. J. Math. 1964. V. 14. P. 559–583.

[4] Batikyan B. T., Gorin E. A., ”Remark on non-local algebras”, Zapiski Nauch. seminar, LOMI (in Russian) 1976. Vol. 65. P. 172–177.

[5] Lang S., Algebra, Addison–Wesley Publishing Company, 1965.

POINT DERIVATIONS 37

Mathematics Institute, Academy of Science, Armenia. E-mail address: bag@instmath.sci.am

Received May, 25, 2000