Haagerup property for C*-algebras and rigidity of C*-algebras with property (T)

We study the Haagerup property for C*-algebras. We first give new examples of C*-algebras with the Haagerup property. A nuclear C*-algebra with a faithful tracial state always has the Haagerup property, and the permanence of the Haagerup property for C*-algebras is established. As a consequence, the class of all C*-algebras with the Haagerup property turns out to be quite large. We then apply Popa's results and show the C*-algebras with property (T) have a certain rigidity property. Unlike the case of von Neumann algebras, for the reduced group C*-algebras of groups with relative property (T), the rigidity property strongly fails in general. Nevertheless, for some groups without nontrivial property (T) subgroups, we show a rigidity property in some cases. As examples, we prove the reduced group C*-algebras of the (non-amenable) affine groups of the affine planes have a rigidity property.Comment: This is the final version. 22pages, no figure

Group C*-algebras as decreasing intersection of nuclear C*-algebras

We prove that for every exact discrete group $\Gamma$, there is an intermediate C*-algebra between the reduced group C*-algebra and the intersection of the group von Neumann algebra and the uniform Roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra $\mathcal{O}_2$. In particular, when $\Gamma$ has the AP (approximation property), the reduced group C*-algebra is realized in this way. We also study extensions of the reduced free group C*-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear C*-algebra is realized in this way.Comment: 22 pages. More detailed proofs. No substantial change. To appear in Amer. J. Math. Expanded version of arXiv:1406.274

Equivariant O2\mathcal{O}_2-absorption theorem for exact groups

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal{O}_2$-absorbing, pointwise outer action on the Cuntz algebra $\mathcal{O}_2$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal{O}_2$-absorption theorem for these groups.Comment: 15 pages, small corrections, to appear in Compositio Mathematic

Chiral multicritical points driven by isospin density in the Ginzburg-Landau approach

We study how a chiral tricritical point (TCP) on QCD phase diagram is affected by the imbalance of up and down quark densities (isospin density), using the generalized Ginzburg-Landau (GL) approach. The resulting phase diagram near TCP shows a rich fine structure which includes inhomogeneities of both the chiral and the charged pion condensations. It turns out that the TCP splits into multicritical points.Comment: 4 pages, 2 eps figures. Presented at QCD@Work 2012: International Workshop on QCD - Theory and Experiment, June 18-21, Lecce (Italy