Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIrreducibility and monicity for representations of $k$-graph $C^*$-algebras
118
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible $Lambda$-semibranching representations and purely atomic representations of $C^*(Lambda)$. Throughout the paper, we work in the setting of row-finite source-free $k$-graphs; this paper constitutes the first analysis of $Lambda$-semibranching representations at this level of generality.
rate research
Read More
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 $Lambda$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.
We initiate the study of real $C^*$-algebras associated to higher-rank graphs $Lambda$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $mathcal{CR}$ $K$-theory of $C^*_{mathbb R} (Lambda, gamma)$ for any involution $gamma$ on $Lambda$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $Lambda$ and the involution $gamma$. We provide a complete description of $K^{CR}(C^*_{mathbb R}(Lambda, gamma))$ for several examples of higher-rank graphs $Lambda$ with involution.
We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish grad
We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by ${0,1}$-valued labellings of their edge sets.
We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
