Abstract
We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many features formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C⁎-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we establish identities when tensoring with strongly self-absorbing C⁎-algebras. Finally, we show how to use the bivariant Cuntz semigroup of the present work to classify unital and stably finite C⁎-algebras.
Highlights
The Cuntz semigroup was introduced in the pioneering work [11] of Joachim Cuntz in the 1970’s as a C∗-analogue of the Murray-von Neumann semigroup of projections in von Neumann algebras, replacing equivalence classes of projections by suitable equivalence classes of positive elements in the union of all matrix iterations of the algebra
We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup
We provide a classification theorem along those lines in Theorem 6.12 (Section 6), where we show that unital stably finite C∗-algebras are isomorphic if and only if there exists a invertible element in the bivariant Cuntz semigroup
Summary
The Cuntz semigroup was introduced in the pioneering work [11] of Joachim Cuntz in the 1970’s as a C∗-analogue of the Murray-von Neumann semigroup of projections in von Neumann algebras, replacing equivalence classes of projections by suitable equivalence classes of positive elements in the union of all matrix iterations of the algebra. We show that every bivariant Cuntz semigroup for a pair of local C∗-algebras is an object of the category W In this categorical framework, continuity, appears to remain a special feature of the ordinary Cuntz semigroup. Following alternative definitions of the ordinary Cuntz semigroup we give a bivariant extension of the Hilbert module picture described in [10] This has the bearings of Kasparov cycles for KK-theory, but with a more suitable set of axioms to accommodate the different nature of the equivalence relation. In order to do so we need to extend some well-known results concerning c.p.c. order zero maps to the setting of local C∗-algebras in the sense of [4], together with some other technical results that are used throughout These are employed to investigate the properties of additivity, functoriality and stability of the bifunctor introduced within this section. K The C∗-algebra of compact operators on a infinite-dimensional separable Hilbert space
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