On the unital C\(^*\)-algebras generated by certain subnormal tuples
For a Hilbert space \({\mathcal H}\), let \({\mathcal B}({\mathcal H})\) denote the algebra of all bounded linear operators on \({\mathcal H}\). An \(m\)-tuple \(S= (S_1,\dots, S_m)\) of commuting operators \(S_i\) in \({\mathcal B}({\mathcal H})\) is said to be subnormal if there exist a Hilbert space \({\mathcal I}\) containing \({\mathcal H}\) and an \(m\)-tuple \(N= (N_1,\dots, N_m)\) of commuting normal operators \(N_i\) in \({\mathcal B}({\mathcal I})\) such that \(N_i{\mathcal H}\subset{\mathcal H}\) and \(N_i|_{\mathcal H}= S_i\) for \(1\leq i\leq m\). The author considers the subnormal \(m\)-tuples \(M_{p}=(M_{p,z_{1}},\dots, M_{p,z_{m}})\), where \( M_{p,z_{i}}\) denotes the operator of multiplication by the coordinate function \(z_{i}\), on a class of reproducing kernel Hilbert spaces \({\mathcal{H}}_{p}\); namely, \({\mathcal{H}}_{m}\) is the Hardy space and \({\mathcal{H}}_{m+1}\) is the Bergman space on the unit ball \(B^{2m}\) of \(\mathbb{C}^m\). These \(m\)-tuples play an important role in the model theory of subnormal operator \(m\)-tuples having their joint Taylor spectra contained in the closure of \(B^{2m}\). Let \(C^*(M_p)\) denote the unital \(C^*\)-algebra generated by \(M_p\) and let \(C^*(\widetilde M_p)\) denote the unital \(C^*\)-algebra generated by the dual \(\widetilde M_p\) of \(M_p\). By using the Brown-Douglas-Fillmore theory and \(K\)-theory, the author proves that \(C^*(M_m)\) and \(C^*(\widetilde M_m)\) are *-isomorphic, but \(C^*(\widetilde M_m)\) and \(C^*(\widetilde M_{m+1})\) are not even homotopy equivalent. In fact, the author investigates such relationships for any two of the \(C^{\ast}\)-algebras in the collection \(\{C^{\ast}(M_{p}), C^{\ast}(\widetilde{M_{p}}): p \geq m\}\).
scientific article; zbMATH DE number 5808428
scientific article