We consider an important class of subnormal operator m-tuples Mp (p = m,m + 1, . . .) that is associated with a class of reproducing kernel Hilbert spaces \documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal H}_p}$$\end{document} (with Mm being the multiplication tuple on the Hardy space of the open unit ball \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb B}^{2m}}$$\end{document} in \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb C}^m}$$\end{document} and Mm+1 being the multiplication tuple on the Bergman space of \documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb B}^{2m}}$$\end{document}). Given any two C*-algebras \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal A}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal B}$$\end{document} from the collection \documentclass[12pt]{minimal}
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\begin{document}$${\{C^*({M}_p), C^*({\tilde M}_p): p \geq m\}}$$\end{document} , where C*(Mp) is the unital C*-algebra generated by Mp and \documentclass[12pt]{minimal}
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\begin{document}$${C^*({\tilde M}_p)}$$\end{document} the unital C*-algebra generated by the dual \documentclass[12pt]{minimal}
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\begin{document}$${{\tilde M}_p}$$\end{document} of Mp, we verify that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal A}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal B}$$\end{document} are either *-isomorphic or that there is no homotopy equivalence between \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal A}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal B}$$\end{document} . For example, while C*(Mm) and C*(Mm+1) are well-known to be *-isomorphic, we find that \documentclass[12pt]{minimal}
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\begin{document}$${C^*({\tilde M}_m)}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${C^*({\tilde M}_{m+1})}$$\end{document} are not even homotopy equivalent; on the other hand, C*(Mm) and \documentclass[12pt]{minimal}
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\begin{document}$${C^*({\tilde M}_{m})}$$\end{document} are indeed *-isomorphic. Our arguments rely on the BDF-theory and K-theory.
