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

Abstract:

Let A be a locally ${B^\ast }$-equivalent Banach $^\ast$-algebra. Then A possesses a unique norm $| \cdot |$ with the property that $|{a^\ast }a| = |a{|^2}$ for all $a \in A$. Let B be the ${B^\ast }$-algebra which is the completion of A in the norm $| \cdot |$. In this paper it is shown that there exists a closed ${B^\ast }$-equivalent $^\ast$-ideal of A which contains the maximal GCR ideal of B. In particular, when B is a GCR algebra, then $A = B$.