Questions concerning matrix algebras and invariance of spectrum

Abstract

Let $A$ and $B$ be unital Banach algebras with $A$ a subalgebra of $B$. Denote the algebra of all $n\times n$ matrices with entries from $A$ by $M_{n}(A)$. In this paper we prove some results concerning the open question: If $A$ is inverse closed in $B$, then is $M_{n}(A)$ inverse closed in $M_{n}(B)$? We also study related questions in the setting where $A$ is a symmetric Banach *-algebra.