Operator ideals and assembly maps in $K$-theory

TL;DR

This paper presents a new proof of Yu's rational injectivity result for the algebraic $K$-theory assembly map related to Schatten ideals, using cyclic homology instead of controlled topology techniques.

Contribution

It offers an alternative proof of Yu's theorem on the injectivity of the assembly map, employing cyclic homology and homotopy $K$-theory, avoiding coarse geometry methods.

Findings

The rational assembly map for homotopy $K$-theory of Schatten ideals is injective.

The new proof aligns with Yu's result by showing the equivalence of assembly maps.

The approach simplifies the proof by using cyclic homology techniques.

Abstract

Let $\cB$ be the ring of bounded operators in a complex, separable Hilbert space. For $p>0$ consider the Schatten ideal $\cL^p$ consisting of those operators whose sequence of singular values is $p$-summable; put $\cS=\bigcup_p\cL^p$. Let $G$ be a group and $\vcyc$ the family of virtually cyclic subgroups. Guoliang Yu proved that the $K$-theory assembly map \[ H_*^G(\cE(G,\vcyc),K(\cS))\to K_*(\cS[G]) \] is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients $\cS$ and the use of some results about algebraic $K$-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu's result. Our proof uses the usual Chern character to cyclic homology. Like Yu's, it relies on results on algebraic $K$-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy $K$-theory. We prove that the rational assembly map \[ H_*^G(\cE(G,\fin),KH(\cL^p))\otimes\Q\to KH_*(\cL^p[G])\otimes\Q \] is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.