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

Abstract

Quillen introduced a new $K_0^{\prime }$-theory of nonunital rings and showed that, under some assumptions (weaker than the existence of unity), this new theory agrees with the usual algebraic $K_0^{alg}$-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 $K_0^{\prime }$ 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 $C^{\ast }$-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 $C^{\ast }$-algebras and some other potential applications to topological field theories are discussed towards the end.

24 pages, the manuscript of an article published in 2011. arXiv admin note: text overlap with arXiv:0906.5400

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