Higher nonunital Quillen K'-theory, KK-dualities and applications to topological $\mathbb{T}$-dualities

TL;DR

This paper introduces a higher nonunital K-theory for k-algebras extending Quillen's original functor, demonstrating its Morita invariance and excision, and applying it to topological T-duality and C*-algebra deformations.

Contribution

It develops a new higher nonunital K-theory extending Quillen's K'_0, proving its key properties and linking it to topological K-theory and T-duality applications.

Findings

KQ-theory agrees with topological K-theory of stable C*-algebras

KQ-theory is Morita invariant and satisfies excision

DG categorical formalism of T-duality using bivariant K-theory

Abstract

Quillen introduced a new $K'_0$-theory of nonunital rings and showed that, under some assumptions (weaker than the existence of unity), this new theory agrees with the usual algebraic $K^{alg}_0$-theory. For a field $k$ of characteristic $0$, we introduce higher nonunital $K$-theory of $k$-algebras, denoted $KQ$, which extends Quillen's original definition of the $K'_0$ functor. We show that the $KQ$-theory is Morita invariant and satisfies excision connectively, in a suitable sense, on the category of idempotent $k$-algebras. Using these two properties we show that the $KQ$-theory agrees with the topological $K$-theory of stable $C^*$-algebras. The machinery enables us to produce a DG categorical formalism of topological homological $\mathbb{T}$-duality using bivariant $K$-theory classes. A connection with strong deformations of $C^*$-algebras and some other potential applications to topological field theories are discussed towards the end.