The Multipliers for Commutative H*-Algebras

Abstract

An H*-algebra is a Banach algebra A with involution* which is a Hilbert space under a scalar product <·,·> such that a) $\left\| x \right\| = \sqrt {\left\langle {x,x} \right\rangle }$ , that is, the Hilbert space norm agrees with the Banach algebra norm, b) $\left\| {{x^*}} \right\| = \left\| x \right\|$c) x* x ≠ 0 if x ≠ 0 and d) <x y,z> = <y, x* z> for all x, y, z∈A. The standard example of an H*-algebra is the algebra L₂(G) for a compact group G with the usual convolution multiplication and scalar product. A general discussion of H*-algebras can be found in Loomis [1] and Naimark [1].