Paschke Categories, K-homology, and the Riemann-Roch Transformation

TL;DR

This paper introduces the Paschke Category for separable C*-algebras, showing its K-theory aligns with topological K-homology, and constructs a Riemann-Roch transformation linking algebraic and topological K-theories of complex manifolds.

Contribution

It defines the Paschke Category, proves its K-theory is isomorphic to topological K-homology, and constructs a Riemann-Roch transformation using Kasparov K-theory techniques.

Findings

Paschke Category's K-theory is isomorphic to topological K-homology.

Constructed an acyclic chain complex inducing a Riemann-Roch transformation.

Established a functorial framework connecting algebraic and topological K-theories.

Abstract

For a separable $C^*$-algebra $A$, we introduce an exact $C^*$-category called the Paschke Category of $A$, which is completely functorial in $A$, and show that its K-theory groups are isomorphic to the topological K-homology groups of the $C^*$-algebra $A$. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation in the homotopy category of spectra, from the algebraic K-theory spectrum of a complex manifold $X$, to its topological K-homology spectrum.