Generalized inverses over Banach algebras
Let \({\mathcal B}\) be a commutative, complex, unital Banach algebra with maximal ideal space \({\mathcal M}\). Let \(A=(a_{ij})\in{\mathcal B}^{n\times m}\) be a matrix with entries in \({\mathcal B}\). Denote by \(\rho(A)\) the rank of \(A\). For \(\varphi\) in \({\mathcal M}\) put \(A\hat{}(\varphi)=(a\hat{}_{ij}(\varphi))\). The author proves that for a given \(A\) there is a \(G\in{\mathcal B}^{m\times n}\) such that (1) \(AGA=A\) and (2) \(GAG=G\) if and only if there is a decomposition \({\mathcal M}=F_ 1\cup\dots\cup F_ k\) onto closed and open subsets such that \(A\hat{}(\varphi)\) has constant rank on each \(F_ i\), and for corresponding idempotents \(e_ i\) (\(e\hat{}_ i=1\) on \(F_ i\) and 0 outside of \(F_ i\)) it is \(\rho(A\hat{}(\varphi))-\rho(e_ i A)\) for \(\varphi\) in \(F_ i\). If, moreover, \(A\) has a symmetric involution then in addition to (1) and (2) \(G\) can be chosen so that \((GA)^*=GA\) and \((AG)^*=AG\).
scientific article; zbMATH DE number 63973