[Paper Review] The generators and relations picture of $KK$-theory

Bernhard Burgstaller|arXiv (Cornell University)|Feb 9, 2016

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper presents a new, elementary generators-and-relations formulation of Kasparov's KK-theory using formal sums and products of *-homomorphisms and synthetic inverses, constructing a category called GK-theory. The key result is that for separable C*-algebras, GK-theory is isomorphic to standard KK-theory, with the Kasparov product naturally arising as composition in the category, offering a simplified, universal definition applicable to broader categories of algebras with actions.

ABSTRACT

This is half an overview article since what we describe here is essentially known. We describe $KK$-theory by generators and relations in a formal sum of formal products of $*$-homomorphisms and some synthetical morphisms. What comes out is a category. The Kasparov product is then just the composition of morphisms. This description may be interesting to anyone who wants a quick and elementary definition of $KK$-theory. The description could also be used for other categories of algebras than $C^*$-algebras endowed with group actions, for example, $C^*$-algebras equipped with an action by a semigroup, a category et cetera.

Motivation & Objective

• To provide a new, elementary, and universally applicable definition of KK-theory using generators and relations.
• To simplify the technical complexity of Kasparov's original construction by embedding the Kasparov product as natural composition in a category.
• To extend the framework beyond C*-algebras with group actions to other categories, such as those with semigroup or category actions.
• To demonstrate that the resulting GK-theory category is isomorphic to standard KK-theory for separable C*-algebras.
• To show that the universal properties of stability, homotopy invariance, and split-exactness naturally emerge from the generators-and-relations construction.

Proposed method

• Define a free category generated by *-homomorphisms and synthetic inverses, forming formal products and sums.
• Introduce equivalence relations on formal morphisms to enforce stability, homotopy invariance, and split-exactness via universal properties.
• Construct GK-theory as the quotient of a free category by relations derived from the universal functorial properties of KK-theory.
• Use Higson’s universal characterization of KK-theory to establish that GK-theory satisfies the same universal property.
• Prove that for separable C*-algebras, GK-theory is isomorphic to standard KK-theory via the uniqueness of the universal functor.
• Extend the framework to other categories of algebras by requiring equivariant morphisms and replacing the compact operators with appropriate stabilizing algebras.

Experimental results

Research questions

• RQ1 Can KK-theory be reconstructed from a simple generators-and-relations presentation using only basic category theory and C*-algebra concepts?
• RQ2 Does the Kasparov product emerge naturally as composition in a category defined by such a presentation?
• RQ3 Is GK-theory isomorphic to standard KK-theory when restricted to separable C*-algebras with group actions?
• RQ4 Can this generators-and-relations approach be generalized to C*-algebras with actions by semigroups, categories, or other structures?
• RQ5 Does the universal property of stability, homotopy invariance, and split-exactness arise intrinsically from the relations in the GK-theory construction?

Key findings

• GK-theory is constructed as a category whose morphisms are formal sums and products of *-homomorphisms and synthetic inverses, modulo relations enforcing stability, homotopy invariance, and split-exactness.
• The Kasparov product is naturally realized as composition of morphisms in GK-theory, eliminating the need for separate technical definition.
• For separable C*-algebras, GK-theory is isomorphic to standard KK-theory, as both satisfy the same universal functorial property.
• The construction allows for a natural descent homomorphism from equivariant KK-theory to ordinary KK-theory via the crossed product construction.
• The framework generalizes beyond group actions to semigroups, categories, and other actions, provided morphisms are equivariant and a stabilizing algebra is used.
• When the cardinality of C*-algebras is bounded, GK-theory morphism classes become sets, and GK-theory is equivalent to a category with actual morphism sets.