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Subalgebras of modular annihilator algebras

1969; Cambridge University Press; Volume: 66; Issue: 1 Linguagem: Inglês

10.1017/s0305004100044637

1469-8064

Bruce A. Barnes,

Advanced Algebra and Logic

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as S B . B is a modular annihilator algebra if B / S B is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B ; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f , continuous on the closed unit disk D in the complex plane, analytic in the interior of D , and such that f (0) = 0. A is a Banach algebra in the usual sup norm over D . Now consider the norm on A defined by Let B be the completion of A in this norm. A has an involution * defined by and also ‖ ff *‖ = ‖ f ‖ 2 for all f ∈ A . Therefore B is a B *-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B . Thus B can be identified with the algebra of all complex sequences which converge to zero.