Cohomology groups of commutative Banach algebras

Abstract

Banach algebras and to investigate some consequences that may be derived from this extension. Suppose A is an associative algebra and Pis an A1-module. Then in [5](2) Hochschild attached to this pair, A, P a sequence of abelian groups Hk(A, P), k = 1, 2, • • •. These groups are called cohomology groups. Three principal results of [5] are the following. Theorem A. In order that the finite dimensional associative algebra A be separable, i.e. semi-simple over every algebraic extension of the ground field, it is necessary and sufficient that Hk(A, P) vanish for all two-sided A-modules P and all positive integers k. Theorem B. If A is a finite dimensional associative algebra, then a necessary and sufficient condition that H2(A, P) should vanish for all A-modules P is that for every finite dimensional algebra B with radical R, if B/R = A, then there exists a subalgebra A' of B with B=A'@R. Theorems A and B are then used to give another proof of the Wedderburn principal theorem. Theorem C (Wedderburn). Let B be a finite dimensional associative algebra and R its radical. If A =73/7? is separable, then there exists a subalgebra A' of B with 73 = A'67?. Without topological considerations if we do not require the algebra A to be finite dimensional, then Theorems A and C are no longer valid. For a discussion of this we refer the reader to [9] and [ll]. When considering the Wedderburn principal theorem for Banach algebras we shall require that the subalgebra A' of 73 be closed and hence that A' is both isomorphic and norm-equivalent to A. In [4], Feldman has shown that there exists a Banach algebra 73 with one-dimensional radical 7? such that 73/7? = /2 (the algebra of square summable sequences with coordinate-wise

Commutativity and ideals in the biduals of topological algebras

“…845], and might be the appropriate answer to his question. The inspiration for our proof comes from Theorem 4.2 of [12]. LEMMA …”

On absolutely continuous functions and the well-bounded operator

“…Then x* is a bounded linear functional on L^ and, therefore, there is a|G 6α{Z, ^f 9 m] such that (3.3.1) x…”

“…Idempotents in this algebra are mapped into projections in J5(X), the algebra of bounded operators on X, by means of a homomorphism extension technique which extends the original operational calculus a->a(T) defined on AC 0 to a bounded homomorphism of AC** into B(X). Kamowitz has used this extension procedure in [9]. When X is reflexive, the extended homomorphism is defined on a quotient algebra of AC**.…”

On Cohomology Groups of Banach Algebras

“…In this note we will study the cohomology groups of some special classes of Banach algebras, and show that there are some relationships between the cohomology groups of a Banach algebra (for definition see [2 ]) and the space of maximal ideals of that Banach algebra We assume that the Banach algebras considered here are the Banach algebras over the field of complex numbers C.…”

“…When E is of type (i), then by Lemma 2, Â is the Banach algebra of all continuous functions defined on E. Let r be any element of E, then the function x^ Ä->C defined by Xr(X) =X(t), for XEÄ, is a continuous algebra homomorphism and Xr(z) =t. When we regard C as a two sided Banach A-module by Xr, then by [2 ] we have 771(Â, C) = 0.…”