On the Unital C*-Algebras Generated by Certain Subnormal Tuples

Abstract

We consider an important class of subnormal operator m-tuples M _p (p = m,m + 1, . . .) that is associated with a class of reproducing kernel Hilbert spaces ${{\mathcal H}_p}$ (with M _m being the multiplication tuple on the Hardy space of the open unit ball ${{\mathbb B}^{2m}}$ in ${{\mathbb C}^m}$ and M _m+1 being the multiplication tuple on the Bergman space of ${{\mathbb B}^{2m}}$). Given any two C*-algebras ${\mathcal A}$ and ${\mathcal B}$ from the collection ${\{C^*({M}_p), C^*({\tilde M}_p): p \geq m\}}$ , where C*(M _p ) is the unital C*-algebra generated by M _p and ${C^*({\tilde M}_p)}$ the unital C*-algebra generated by the dual ${{\tilde M}_p}$ of M _p , we verify that ${\mathcal A}$ and ${\mathcal B}$ are either *-isomorphic or that there is no homotopy equivalence between ${\mathcal A}$ and ${\mathcal B}$ . For example, while C*(M _m ) and C*(M _m+1) are well-known to be *-isomorphic, we find that ${C^*({\tilde M}_m)}$ and ${C^*({\tilde M}_{m+1})}$ are not even homotopy equivalent; on the other hand, C*(M _m ) and ${C^*({\tilde M}_{m})}$ are indeed *-isomorphic. Our arguments rely on the BDF-theory and K-theory.