Locally $B^{\ast }$-equivalent algebras

Abstract:

Let A be a Banach $^ \ast$-algebra. A is locally ${B^ \ast }$-equivalent if, for every selfadjoint element $t \in A$, the closed $^ \ast$-subalgebra of A generated by t is $^\ast$-isomorphic to a ${B^ \ast }$-algebra. In this paper it is shown that when A is locally ${B^\ast }$-equivalent, and in addition every selfadjoint element in A has at most countable spectrum, then A is $^ \ast$-isomorphic to a ${B^ \ast }$-algebra.