Let $Gamma$ be a discrete group freely acting via homeomorphisms on the compact Hausdorff space $X$ and let $C(X)rtimes_eta Gamma$ be the completion of the convolution algebra $C_c(Gamma, C(X))$ with respect to a $C^*$-norm $eta$. A non-zero ideal $J
unlhd C(X)rtimes_etaGamma$ is exotic if $Jcap C(X) =(0)$. We give examples showing that exotic ideals are present whenever $Gamma$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide negative answers to two questions of K. Thomsen. Let $mathfrak{A}$ be a non-atomic MASA on a separable Hilbert space and let $mathcal B_0$ be the linear span of the unitary operators $Uinmathcal B(mathcal H)$ such that $Umathfrak{A} U^*=mathfrak{A}$. We observe that while $mathcal B_0$ contains no compact operators, the norm-closure of $mathcal B_0$ contains all compact operators. This gives a positive answer to a question of A. Katavolos and V. Paulsen. For a free action of $Gamma$ on a compact Hausdorff space $X$, the opaque and grey ideals in $C(X)rtimes_eta Gamma$ coincide. We conclude with an example of a free action of $Gamma$ on a compact Hausdorff space $X$ along with a norm $eta$ for which these ideals are non-trivial.
We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1
051-1082 (2004). We show only the uniqueness of the beta-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $mu$) and its relation with the eigen-probability $ u_beta$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for the Gibbs state. There is no Phase transition for the KMS state.
Given a graph $E$, an action of a group $G$ on $E$, and a $G$-valued cocycle $phi$ on the edges of $E$, we define a C*-algebra denoted ${cal O}_{G,E}$, which is shown to be isomorphic to the tight C*-algebra associated to a certain inverse semigroup
$S_{G,E}$ built naturally from the triple $(G,E,phi)$. As a tight C*-algebra, ${cal O}_{G,E}$ is also isomorphic to the full C*-algebra of a naturally occurring groupoid ${cal G}_{tight}(S_{G,E})$. We then study the relationship between properties of the action, of the groupoid and of the C*-algebra, with an emphasis on situations in which ${cal O}_{G,E}$ is a Kirchberg algebra. Our main applications are to Katsura algebras and to certain algebras constructed by Nekrashevych from self-similar groups. These two classes of C*-algebras are shown to be special cases of our ${cal O}_{G,E}$, and many of their known properties are shown to follow from our general theory.
We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set
$S$ and a stationary continuous time Markov Chain $X_t$, $tgeq 0$, taking values on S. We denote by $Omega$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($Omega, {cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${cal L}^infty (P)$. From the probability $P$ we define a Ruelle operator ${cal L}^t, tgeq 0$, acting on functions $f:Omega to mathbb{R}$ of ${cal L}^infty (P)$. Given $V:Omega to mathbb{R}$, such that is constant in sets of the form ${X_0=c}$, we define a modified Ruelle operator $tilde{{cal L}}_V^t, tgeq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ u_V$ on $Omega$ associated to $tilde{{cal L}}^t_V, tgeq 0$. We also show the following property for the probability $ u_V$: for any integrable $gin {cal L}^infty (P)$ and any real and positive $t$ $$ int e^{-int_0^t (V circ Theta_s)(.) ds} [ (tilde{{cal L}}^t_V (g)) circ theta_t ] d u_V = int g d u_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).
