Morita invariance of unbounded bivariant K-theory

TL;DR

This paper develops a Morita invariance framework for unbounded bivariant K-theory of operator *-algebras, establishing isomorphisms and refinements of Kasparov products, with applications to various algebraic structures.

Contribution

It introduces Morita equivalence for operator *-algebras and connects unbounded bivariant K-theory with Kasparov's theory, providing new invariance and refinement results.

Findings

Unbounded Kasparov product induces isomorphisms between twisted spectral triples over Morita equivalent algebras.

Unbounded bivariant K-theory relates to Kasparov's K-theory via the Baaj-Julg transform.

C^1-versions of Morita equivalences demonstrated in hereditary subalgebras, conformal, and crossed product contexts.

Abstract

We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C^1-versions of well-known C^*-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.