Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAll finite transitive graphs admit self-adjoint free semigroupoid algebras
96
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is $B(mathcal{H})$. This is accomplished through a new construction that reduces this problem to in-degree $2$-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of Beal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras.
rate research
Read More
A free semigroupoid algebra is the closure of the algebra generated by a TCK family of a graph in the weak operator topology. We obtain a structure theory for these algebras analogous to that of free semigroup algebra. We clarify the role of absolute continuity and wandering vectors. These results are applied to obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with reflexivity, a Kaplansky density theorem and classification for free semigroupoid algebras. Several classes of examples are discussed and developed, including self-adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence.
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.
Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous with a density that is emph{constant along the vertical direction} in the support of $rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $rho_t$ by a natural map is the law of $x_0+s_t$. Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form [frac{1}{r^2}w_t(theta)] in polar coordinates in its support. The support of $mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $mu_t$ by a natural map is the law of $uu_t$. In the special case that $u$ is Haar unitary, the Brown measure $mu_t$ follows the emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by [frac{1}{2pi t}frac{1}{r^2}] in its support.
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
In this paper we consider a bootstrap class $mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Gammainmathfrak C$ and any strongly self-absorbing C*-algebra $mathcal D$: (1) There is a unique strongly outer $Gamma$-action on $mathcal D$ up to (very strong) cocycle conjugacy. (2) If $alpha: Gammacurvearrowright A$ is a strongly outer action on a separable, unital, nuclear, simple, $mathcal D$-stable C*-algebra with at most one trace, then it absorbs every $Gamma$-action on $mathcal D$ up to (very strong) cocycle conjugacy. In fact we establish more general relati
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
