相关论文: Higher rank graphs, k-subshifts and k-automata

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph $\Lambda$ is a natural generalization of a higher rank graph. A pullback of $\Lambda$ is constructed by pulling it back over a given monoid morphism to $P$, while a…

Higher rank graphs, also known as $k$-graphs, are a $k$-dimensional generalization of directed graphs and a rich source of examples of $C^*$-algebras. In the present paper, we contribute to the geometric classification program for $k$-graph…

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…

We give a local treatment of finite alignment by identifying the finitely aligned part of any (not necessarily finitely aligned) higher-rank graph. We show the finitely aligned part is itself a constellation and forms a finitely aligned…

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…

A well-known result of Brylawski constructs an elementary lift of a matroid $M$ from a linear class of circuits of $M$. We generalize this result by showing how to construct a rank-$k$ lift of $M$ from a rank-$k$ matroid on the set of…

Let $G$ be a connected complex semisimple Lie group, $\Gamma$ be a cocompact, irreducible and torsionless lattice in $G$ and $K$ be a maximal compact subgroup of $G$. Assume $\Gamma$ acts by left multiplication and $K$ acts by right…

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in N. We focus on semigroups P arising as part of a quasi-lattice ordered group (G,P) in the sense of…

We introduce the notion of a self-similar action of a groupoid $G$ on a finite higher-rank graph. To these actions we associate a compactly aligned product system of Hilbert bimodules, and thereby obtain corresponding universal…

It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. It has been shown that such patterns can occur when the alphabet is…

Given a row-finite higher-rank $k$-graph $\Lambda$, we define a commutative monoid $T_\Lambda$ which is a higher-rank analogue of the talented monoid of a directed graph. The talented monoid $T_\Lambda$ is canonically a…

Let Gamma be a semidirect product of the form Z^n rtimes Z/p where p is prime and the Z/p-action on Z^n is free away from the origin. We will compute the topological K-theory of the real and complex group C*-algebra of Gamma and show that…

Subshifts with property $(A)$ are constructed from a class of directed graphs. As special cases the Markov-Dyck shifts are shown to have property $(A)$. The semigroups, that are associated to $\mathcal R$-graph shifts with Property (A), are…

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…

We show that every topological k-graph constructed from a locally compact Hausdorff space {\Omega} and a family of pairwise commuting local homeomorphisms on {\Omega} satisfying a uniform boundedness condition on the cardinalities of…

Given a row-finite, source-free, graph of rank k, we extend the definition of reduction introduced by Eckhardt et al. This constitutes a large step forward in the extension of the geometric classification of finite directed graph…

A hyperfinite $II_1$ subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusion…

Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a…

We consider non-selfadjoint operator algebras $\mathfrak{L}(G,\lambda)$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of…

Given a non-trivial complete valued field $K$ with value group $\Lambda$, we construct a $\Lambda$-tree space associated to $K$ analog of the Bruhat-Tits tree, and locally finite trees associated to compact subsets of the projective line.…