Related papers: Separable representations, KMS states, and wavelet…

In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $\Lambda$. We begin by giving an alternative characterization of the…

In this monograph we undertake a comprehensive study of separable representations (as well as their unitary equivalence classes) of $C^*$-algebras associated to strongly connected finite $k$-graphs $\Lambda$. We begin with the…

The representations of a $k$-graph $C^*$-algebra $C^*(\Lambda)$ which arise from $\Lambda$-semibranching function systems are closely linked to the dynamics of the $k$-graph $\Lambda$. In this paper, we undertake a systematic analysis of…

In this paper we show how to produce a large number of representations of a graph C*-algebra in the space of the bounded linear operators in $L^2(X,\mu)$. These representations are very concrete and, in the case of graphs that satisfy…

In recent joint work of the authors with Laca, we precisely formulated the notion of partition function in the context of C*-dynamical systems. Here, we compute the partition functions of C*-dynamical systems arising from Toeplitz algebras…

For a finite, strongly connected $k$-graph $\Lambda$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^*$-algebra $C^*(\Lambda)$. They found that these KMS states are determined…

In this article we show that there are branching systems (which induce representations of the graph algebra $C^*(E)$) associated to each row-countable graph $E$. For row-countable graphs, we characterize the condition $(L)$ via branching…

We construct a locally compact Hausdorff topology on the path space of a finitely aligned $k$-graph $\Lambda$. We identify the boundary-path space $\partial\Lambda$ as the spectrum of a commutative $C^*$-subalgebra $D_\Lambda$ of…

We study the phase transition of KMS states for the C*-algebras of $ax+b$-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz,…

Let $G$ be a countable discrete amenable group, and $\Lambda$ be a strongly connected finite $k$-graph. If $(G,\Lambda)$ is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the…

The Toeplitz algebra $\mathcal{T}C^{*}(\Lambda)$ for a finite $k$-graph $\Lambda$ is equipped with a continuous one-parameter group $\alpha^{r}$ for each $ r\in \mathbb{R}^{k}$, obtained by composing the map $\mathbb{R} \ni t \to…

We define the notion of a $\Lambda$-system of $C^*$-correspondences associated to a higher-rank graph $\Lambda$. Roughly speaking, such a system assigns to each vertex of $\Lambda$ a $C^*$-algebra, and to each path in $\Lambda$ a…

Given a $k$-graph $\Lambda $ we construct a Markov space $M_\Lambda $, and a collection of $k$ pairwise commuting cellular automata on $M_\Lambda $, providing for a factorization of Markov's shift. Iterating these maps we obtain an action…

We produce a complete descrption of the lattice of gauge-invariant ideals in $C^*(\Lambda)$ for a finitely aligned $k$-graph $\Lambda$. We provide a condition on $\Lambda$ under which every ideal is gauge-invariant. We give conditions on…

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects…

We study the equilibrium or KMS states of the Toeplitz C*-algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of…

We define branching systems for finitely aligned higher-rank graphs. From these we construct concrete representations of higher-rank graph C*-algebras on Hilbert spaces. We prove a generalized Cuntz-Krieger uniqueness theorem for periodic…

Given a $k$-graph $\Lambda$ and an element $p$ of $\NN^k$, we define the dual $k$-graph, $p\Lambda$. We show that when $\Lambda$ is row-finite and has no sources, the $C^*$-algebras $C^*(\Lambda)$ and $C^*(p\Lambda)$ coincide. We use this…

The relative graph $C^*$-algebras introduced by Muhly and Tomforde are generalizations of both graph algebras and their Toeplitz extensions. For an arbitrary graph $E$ and a subset $R$ of the set of regular vertices of $E$ we show that the…

We use the boundary-path space of a finitely-aligned k-graph \Lambda to construct a compactly-aligned product system X, and we show that the graph algebra C^*(\Lambda) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting,…