A set of maps from \(K\) to \(\mathrm{End}_ A(\ell _ 2(A))\) isomorphic to \(\mathrm{End}_{A(K)}(\ell _ 2(A(K)))\). Applications (Q1067174)

Let \(A\) be a \(C^*\)-algebra, \(K\) be a compact space, \(A(K)\) be the \(C^*\)- algebra of all continuous maps from \(K\) into \(A\), \(\ell_ 2(A)\) be the Hilbert \(A\)-module being standard for all countably generated Hilbert \(A\)- modules in the sense of Kasparov's stabilization theorem. We investigate a set of maps from \(K\) into \(\mathrm{End}_ A(\ell_ 2(A))\) which is isomorphic to \(\mathrm{End}_{A(K)}(\ell_ 2(A(K))).\) We describe the subsets which are isomorphic to \(\mathrm{End}_{A(K)}(\ell_ 2(A(K))),\) \(\mathrm{GL}_{A(K)}(\ell_ 2(A(K)))\) and \(\mathrm{GL}_{A(K)}(\ell_ 2(A(K))),\) respectively. As an application we deduce a criterion for the self-duality of \(\ell_ 2(A)\) for \(A\) being commutative.