A sharpening of the Berkson-Glickfeld theorem

Abstract

It is known that if in a Banach*-algebra with unit the following holds:then it is a C*-algebra (see [3]).We shall show that the above theorem can be sharpened in the following way: we replace the submultiplicativity of the norm by the weaker assumptionObserve that under this assumption even the existence of exp(ih) is not at all obvious, but it will be proved to be true below. Our main result is Theorem 2 which depends on Theorem 1. Our last remark is the equivalent-norm-version of the statement.