On restricted perturbations in inverse images and a description of normalizer algebras in \(C^*\)-algebras
Let \(E\) and \(F\) be Banach spaces and \(T\) a bounded linear map from \(E\) into \(F\). Using a certain perturbation function \(f(\cdot, T)\) the author finds a useful sufficient criterion that \(T\) maps the closed unit ball onto a closed set. Applying this to the quotient map from a \(C^*\)- algebra \(A\) onto its quotient Banach space \(A/ (L+ R)\) by the sum \(L+R\) of closed left and right ideals \(L\) and \(R\) of \(A\) he obtains that the closed unit ball of \(A\) maps onto the closed unit ball of \(A/ (L+ R)\). This gives an independent description of the images of the right normalizer algebra \(N_r (D)\) and the normalizer algebra \(N(D)= N_l (D)\cap N_r (D)\) of a hereditary \(C^*\)-subalgebra \(D\) of \(A\) by the quotient map from \(A\) onto \(A/(\text{cl} (AD)+ \text{cl} (DA))\). Then he uses this to prove a necessary and sufficient criterion for a \(C^*\)- algebra \(B\) to be a \(C^*\)-quotient-algebra of a \(C^*\)-subalgebra of a \(C^*\)-algebra \(A\). The latter criterion will be applied in the situation where \(A\) equals the CAR-algebra in some forthcoming papers.
scientific article; zbMATH DE number 764016