# “Operator K-theory” has appeared on AMS Open Math Notes

My final Penn State course (Spring 2017) was about K-theory and operator algebras – the connection between these two has been central to my mathematical life.  I wrote up lecture notes for this course, as has become usual for me.  I’m pleased to report that these have now appeared on the AMS Open Math Notes page.

The American Mathematical Society hosts AMS Open Math Notes,  which is “a repository of freely downloadable mathematical works in progress hosted by the AMS as a service to researchers, teachers and students.”

The Open Math Notes homepage continues  “These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision.  Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.”

# Matt Wiersma on exotic group C*-algebras

Recently Matt Wiersma from Waterloo spoke in our seminar about some of his work related to “exotic group C*-algebras”.  A more detailed account is on the arXiv.  I thought I would try to write up some of what I learned (probably, as usual, this is the most elementary points, but it was new to me).

What is an exotic group C*-algebra?  It is a completion of the group algebra which is different from the two standard examples (maximal and reduced) that we describe in C*-algebra courses.  Oversimplifying, we might make an analogy with compactifications of a locally compact Hausdorff space.   There is always a minimal one (one-point compactification) and a maximal (Stone-Cech), but there are also plenty of other things in between.  Analogously, in the case where a group $$\Gamma$$ is non amenable, one might imagine that there should be many other C*-completions of $${\mathbb C}\Gamma$$ lying between the maximal and the reduced C*-algebras.   (Whether, in fact, there exists any group for which $${\mathbb C}\Gamma$$ has exactly two distinct completions appears to be an open question.)

# Paper on sheaves of C*-algebras and K-homology published

After a busy day giving final exams in the MASS program it was nice to learn today that my paper with Paul Siegel about sheaves of C*-algebras has appeared in the Journal of K-Theory.  The link for the published version is

http://journals.cambridge.org/repo_A91rlWyM

This paper arose from some discussions when Paul was writing his thesis.  We were talking about the “lifting and controlling” arguments for Paschke duals that are used in the construction of various forms of operator-algebraic assembly maps (an early example is the one that appears in my paper with Nigel on the coarse Baum-Connes conjecture, which asserts that the quotient $$D^*(X)/C^*(X)$$ of the “controlled” pseudolocal by the “controlled” locally compact operators does not depend on the assumed “control”).  At some point in these discussions I casually remarked that, “of course”, what is really going on is that the Paschke dual is a sheaf.  Some time later I realized that what I had said was, in fact, true.  There aren’t any new results here but I hope that there is some conceptual clarification.   (There is an interesting spectral sequence that I’ll try to write about another time, though.)

# “Finite part of operator K-theory” I

First of all, I apologize for the hiatus in posting over the past couple of weeks,  Organizing a (non-mathematical) conference has absorbed a big chunk of my time, and then getting back up to speed with routine tasks has absorbed another big chunk.   However…

So I started looking at the recent paper of Shmuel Weinberger and Guoliang Yu,  They are interested in looking at the part of the $$K$$-theory of the maximal C*-algebra of a group $$\Gamma$$ which is generated by the projections

$p_H = \frac{1}{|H|} \sum_{h\in H} h\quad \in {\mathbb C}[G]$

in the complex group algebra of $$G$$, where $$H$$ is a finite cyclic subgroup.   (Question: Why do they restrict attention to finite cyclic subgroups? Wouldn’t any finite subgroup work just as well.)

The claim is that these generate a “large” subgroup of $$K_0(C^*_{max}(G))$$ which is not in the image of the maximal assembly map from $$K_0(BG)$$.  “Large” is expressed in terms of a lower bound for the rank of this abelian group.

The basic strategy, so far as I understand it, can be thought of in terms of a familiar argument for property T groups.  Let $$G$$ be any group.  The maximal group C*-algebra has a homomorphism $$\alpha$$ to $$\mathbb C$$, which just is the regular representation (as a linear map on $${\mathbb C}[G]$$ it sends every group element to 1.   On the other hand, the reduced (and therefore also the maximal) group $$C^*$$ algebras have a different trace $$\tau$$ which sends the identity element to 1 and every other element of $$G$$ to 0 – this is the tracial vector state associated to the unit vector $$\xi_e$$ in the regular representation $$\ell^2(G)$$.  At the level of K-theory we get a diagram

$\begin{array}{ccc} K_0(C^*_{max}(G))&\to^\alpha &{\mathbb Z}\\ \downarrow&&\downarrow\\ K_0(C^*_r(G)&\to^\tau & {\mathbb R}\end{array}$

This diagram need not commute.  In fact, if $$G$$ has property T and we consider at the top left corner the K-theory class of the Kazhdan projection – the projection (whose existence is guaranteed by property T) which maps, under any representation, to the projection onto the G-invariant subspace of that representation – then this class maps to 1 by traversing the diagram via the top right corner and to 0 traversing via the lower left corner.   However, it must commute for any element in the image of the (maximal) assembly map, as follows essentially from Atiyah’s $$L^2$$ index theorem.  Thus, as is well known, we infer that the class of the Kazhdan projection is not in the image of the maximal assembly map.

Weinberger and Yu point out that a similar argument can be applied to the projection $$p_H$$ associated to a finite cyclic subgroup $$H$$ of $$G$$. In fact, the homomorphism $$\alpha$$ takes $$[p_H]$$ to 1, whereas the trace $$\tau$$ takes it to $$|H|^{-1}$$.  This is independent of any property T considerations.  Motivated by this, they conjecture that the rank of the subgroup of $$K_0(C^*_{max}(G)))$$ generated by the $$[p_H]$$ (they call this the “finite part” of this group) is at least equal to the number of distinct orders of cyclic subgroups of $$G$$, and that no non-identity element in the finite part lies in the image of the assembly map.

Next time I hope to talk about their approach to proving this in  certain cases.

Weinberger, Shmuel, and Guoliang Yu. Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds. ArXiv e-print, August 21, 2013. http://arxiv.org/abs/1308.4744.