Symmetric Banach \(^*\)-algebras: invariance of spectrum (Q2717547)

When \(A\) is a symmetric unital Banach *-algebra which is a subalgebra of a unital Banach algebra \(B\), the author investigates the relationships among the following three concepts:NEWLINENEWLINENEWLINE(i) \(A\) is inverse closed in \(B\) iff whenever at \(a\in A\) and \(a^{-1}\in B\), \(a^{-1}\in A\).NEWLINENEWLINENEWLINE(ii) \(A\) is *-inverse closed in \(B\) iff whenever \(a^*= a\in A\) and \(a^{-1}\in B\), \(a^{-1}\in A\).NEWLINENEWLINENEWLINE(iii) \(A\) is spectral radius preserving iff \(r(a,A)= r(a,B)\) for all \(a\in A\), where \(r(a,A)\) and \(r(a,B)\) denote the spectral radius of a relative to \(A\) and \(B\), respectively.NEWLINENEWLINENEWLINEClearly, the implications \(\text{(i)}\Rightarrow \text{(ii)}\) and \(\text{(i)}\Rightarrow \text{(iii)}\) hold. JFor the converse implication \(\text{(iii)}\Rightarrow \text{(i)}\) he obtains the result: If \(r*(a^*a, A)= r(a^* a,B)\) for all \(a\in A\), then \(A\) is inverse closed in \(B\). Furthermore, he shows that if the involution on \(A\) is continuous with respect to the \(B\)-norm, then (i), (ii) and (iii) are equivalent. This implies that if \(A\) is a \(C^*\)-algebra and the involution on \(A\) is continuous with respect to the \(B\)-norm, then \(A\) is inverse closed in \(B\).