Completely bounded subcontexts of a Morita context of unital $C^*$-algebras

TL;DR

This paper investigates topological invariants for completely bounded Morita equivalences of holomorphic cross-section algebras, generalizing previous examples from annuli to bordered Riemann surfaces and providing criteria for factorization of such equivalences.

Contribution

It introduces a generalization of invariants for Morita equivalences from annuli to bordered Riemann surfaces and develops criteria for factoring these equivalences into similarity and strong Morita components.

Findings

Estimated norms using conformal invariants of the annulus.

Generalized to bordered Riemann surfaces.

Provided criteria for factorization of Morita equivalences.

Abstract

In this paper, we answer a question of Blecher-Muhly-Paulsen pertaining to identifying topological invariants for completely bounded Morita equivalences of holomorphic cross-section algebras. Given a certain natural subcontext of a strong Morita context of $n$-homogeneous $C^*$-algebras whose spectrum $T$ is an annulus, Blecher-Muhly-Paulsen are able to estimate the norm of a lifting of the identity of a holomorphic subalgebra by a conformal invariant of the annulus and a property of the associated matrix bundle. We give a generalization of the above example in which $T$ is a bordered Riemann surface. While constructing this generalization, we develop a sufficient criterion for when a unital completely bounded Morita equivalence can be factored into a similarity and a strong Morita equivalence.