[Paper Review] Additive C*-categories and K-theory

Ulrich Bunke, Alexander Engel|arXiv (Cornell University)|Oct 28, 2020

Homotopy and Cohomology in Algebraic Topology23 references4 citations

TL;DR

This paper establishes foundational tools for topological K-theory in C*-categories by introducing orthogonal sums of infinite families of objects, constructing reduced crossed products with groups, and axiomatizing K-theory via homological functors. It proves rigidity properties of K-theory and applies them to construct equivariant homology theories on orbit categories, enabling applications in equivariant Kasparov theory and the Baum–Connes conjecture.

ABSTRACT

We review the notions of a multiplier category and the $W^{*}$-envelope of a $C^{*}$-category. We then consider the notion of an orthogonal sum of a (possibly infinite) family of objects in a $C^{*}$-category. Furthermore, we construct reduced crossed products of $C^{*}$-categories with groups. We axiomatize the basic properties of the $K$-theory for $C^{*}$-categories in the notion of a homological functor. We then study various rigidity properties of homological functors in general, and special additional features of the $K$-theory of $C^{*}$-categories. As an application we construct and study interesting functors on the orbit category of a group from $C^{*}$-categorical data.

Motivation & Objective

• To provide a comprehensive framework for topological K-theory in C*-categories, focusing on infinite orthogonal sums and reduced crossed products.
• To axiomatize K-theory for C*-categories using the concept of homological functors, capturing its essential structural properties.
• To establish rigidity results for homological functors and K-theory, particularly in the context of C*-categories with group actions.
• To construct and study functors on the orbit category of a group using C*-categorical data, enabling applications to equivariant homotopy theory.
• To supply essential background for subsequent work on equivariant coarse homology, KK-theory, and Paschke duality in the context of the Baum–Connes conjecture.

Proposed method

• Introduces orthogonal sums of families of objects in C*-categories via a universal property, generalizing Hilbert C*-module direct sums.
• Constructs the reduced crossed product of a C*-category with a group action using a norm derived from a representation on a category of the form $\mathbf{L}^2(G,\mathbf{C})$.
• Defines the multiplier C*-category and relates it to the $W^*$-envelope, extending unitary equivalence to non-unital C*-categories.
• Axiomatizes K-theory as a homological functor, identifying its rigidity and Morita invariance properties.
• Applies Yoneda-type embeddings and left Kan extensions to lift functors from orbit categories to equivariant topological spaces.
• Uses weakly equivariant functors and induction functors $i_G^K$ to construct $K$-equivariant homology theories from $G$-equivariant data.

Experimental results

Research questions

• RQ1 How can orthogonal sums of infinite families of objects be rigorously defined and characterized in C*-categories?
• RQ2 What is the structure of the reduced crossed product of a C*-category with a group action, and how does it relate to representations on $\mathbf{L}^2(G,\mathbf{C})$?
• RQ3 What are the rigidity properties of K-theory when viewed as a homological functor on C*-categories?
• RQ4 How can functors on the orbit category of a group be constructed and extended to equivariant homology theories?
• RQ5 In what way do these constructions support the development of equivariant Kasparov theory and Paschke duality?

Key findings

• The paper constructs a well-defined notion of orthogonal sum for possibly infinite families of objects in unital C*-categories via universal properties.
• It establishes that the reduced crossed product of a C*-category with a group action exists and is characterized by a norm derived from a representation on $\mathbf{L}^2(G,\mathbf{C})$.
• The $W^*$-envelope of a C*-category is constructed and shown to be a key tool in extending unitary equivalence to non-unital settings.
• The K-theory of C*-categories is fully axiomatized as a homological functor, with rigidity and Morita invariance properties proven.
• Functors on the orbit category $G\mathbf{Orb}$ are extended to $K$-equivariant homology theories via left Kan extension, with $E^K(X) \simeq E^G(\mathrm{Res}^K_G(X))$.
• The construction provides the necessary categorical and homotopical foundations for subsequent applications in equivariant coarse homology and the Baum–Connes conjecture.