My C*-algebra notes (from the Fall 2015 course) are now on *AMS Open Math Notes*. You can find them here.

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My C*-algebra notes (from the Fall 2015 course) are now on *AMS Open Math Notes*. You can find them here.

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.)

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.)

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.

In our *Mapping surgery to analysis* papers, Nigel and I proposed an analytic counterpart of the *surgery exact sequence* which summarizes the main results of the (Browder, Novikov, Sullivan, Wall) theory of high-dimensional manifolds. This exact sequence identifies the set of manifold structures within a given homotopy type \(X\) (the *structure set*) as the fiber of an assembly map

\[ H_*(X; {\mathbb L}(e)) \to L_*({\mathbb Z}\pi_1(X)) \]

which abstracts the ides of obtaining “signature obstructions” from a “surgery problem”.

Analogously, we constructed an *analytic structure set* (actually the K-theory of a certain C*-algebra) as the fiber of a Baum-Connes type assembly map, and showed that index theory provides a natural transformation from the topological surgery exact sequence to our *analytic surgery exact sequence*.

Our structure set is defined in purely analytic terms. However, in a subsequent paper where we related our exact sequence to the theory of \(\eta\) invariants, it became useful to have a more geometrical approach to the structure set also. (The relation between the “more geometrical” and “more analytical” approaches is roughly the same as that between the Baum-Douglas and Kasparov models of K-homology.) Our paper didn’t give a geometric *definition* of the structure set – just a geometric *approach* to certain elements.

A recent arXiv paper by Deeley and Goffeng proposes to take this idea to its logical conclusion by constructing a Baum-Douglas type model for the whole analytic structure set. The basic idea is this: An element of the structure set should be “an elliptic operator together with a reason that its index vanishes”. The cobordism invariance of the index shows that one example of such a “reason” is that our elliptic operator is actually defined on the *boundary* of some manifold (and that our operator is a boundary operator). Therefore a first approximation to a Baum-Douglas model of the structure set should have as cycles spin-c manifolds with boundary \( (M,\partial M) \) together with maps \(\partial M \to X\).

But of course this (cobordism) is not the *only* known reason for the vanishing of an index (e.g., as I understand it, the fundamental question about positive scalar curvature metrics is whether positive scalar curvature *implies* some bordism condition). So suppose you have an elliptic operator whose index vanishes for some “positive scalar curvature type” reason. How are you to build a structure class? It seems to me that Deeley-Goffeng deal with this by incorporating quite a lot of analysis into their geometric cycles – as well as the bordism that I have described, there are also projective module bundles over the group algebra, etc… this makes the desired exactness true, but perhaps at the cost of making the groups less geometrical; they are a “geometry-analysis hybrid”. And that is inevitable in this problem.

I should mention that several other applications of the analytic surgery sequence depend on constructing an appropriate ncie model for the structure set: e.g. Siegel, Xie-Yu (see below). I’m not sure whether our original model is “nice” for anybody!

Deeley, Robin, and Magnus Goffeng. *Realizing the Analytic Surgery Group of Higson and Roe Geometrically, Part I: The Geometric Model*. ArXiv e-print, August 27, 2013. http://arxiv.org/abs/1308.5990.

Higson, Nigel, and John Roe. “Mapping Surgery to Analysis. I. Analytic Signatures.” *K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences* 33, no. 4 (2005): 277–299. doi:10.1007/s10977-005-1561-8.

———. “Mapping Surgery to Analysis. II. Geometric Signatures.” *K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences* 33, no. 4 (2005): 301–324. doi:10.1007/s10977-005-1559-2.

———. “Mapping Surgery to Analysis. III. Exact Sequences.” *K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences* 33, no. 4 (2005): 325–346. doi:10.1007/s10977-005-1554-7.

Higson, Nigel, and John Roe. “\(K\)-homology, Assembly and Rigidity Theorems for Relative Eta Invariants.” *Pure and Applied Mathematics Quarterly* 6, no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer (2010): 555–601.

Siegel, Paul. “The Mayer-Vietoris Sequence for the Analytic Structure Group.” *arXiv:1212.0241* (December 2, 2012). http://arxiv.org/abs/1212.0241.

Siegel, Paul. “Homological Calculations with the Analytic Structure Group.” PhD Thesis, Penn State, 2012. https://etda.libraries.psu.edu/paper/16113/.

Xie, Zhizhang, and Guoliang Yu. “A Relative Higher Index Theorem, Diffeomorphisms and Positive Scalar Curvature.” *arXiv:1204.3664* (April 16, 2012). http://arxiv.org/abs/1204.3664.

Xie, Zhizhang, and Guoliang Yu. “Positive Scalar Curvature, Higher Rho Invariants and Localization Algebras.” *arXiv:1302.4418* (February 18, 2013). http://arxiv.org/abs/1302.4418.