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Let T be a compact endomorphism of a commutative semisimple Banach algebra B. This paper discusses the behavior of the adjoint T* of T on the set X / of multiplicative linear functionals on B. In particular it is shown that Π T* n (X ;) is finite, thus generalizing the example of compact endomorphisms of the disc algebra.
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topological considerations if we do not require the algebra A to be finite dimensional, then Theorems A and C are no longer valid.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
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