Higher nonunital Quillen K′-theory, KK-dualities and applications to topological T-dualities

Quillen introduced a new K0′-theory of nonunital rings in Quillen (1996) [1] and showed that, under some assumptions (weaker than the existence of unity), this new theory agrees with the usual algebraic K0alg-theory. For a field k of characteristic 0, we introduce higher nonunital K-theory of k-algebras, denoted as KQ, which extends Quillen's original definition of the K0′ 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 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.