SEMISIMPLE ELEMENTS IN GROUP-ALGEBRAS

Let A be a C-algebra with 1. Let xi = lambda1e1 + ... + lambda(n)e(n) is-an-element-of A where each lambda(i) is-an-element-of C and e1,..., e(n) are orthogonal idempotents in A. We give conditions on A to guara ntee that a certain function of xi is bounded in terms of the lambda(i )'s. Examples of such algebras are the complex group ring C[G] and the ring M(m)(C[G]) of matrices over C[G], for arbitrary groups G. Thus t his paper generalizes the result of [PP] on C[G] (without using C-alg ebra theory), and that of [LP] on M(m)(C[G]) (without assuming that G is finite).