arXiv subjects

Menassie Ephrem

Publications and source records attributed to Menassie Ephrem.

5 recordsLinked to original sources

arXiv2020

On Labeled Graph $C^*$-algebras

Given a directed graph $E$ and a labeling $\mathcal{L}$, one forms the labeled graph $C^*$-algebra by taking a weakly left--resolving labeled space $(E, \mathcal{L}, \mathcal{B})$ and considering a universal generating family of partial isometries and projections. In this paper, we work on ideals for a labeled graph $C^*$-algebra when the graph contains sinks. Using some of the tools we build, we compute $C^*(E, \mathcal{L}, \mathcal{B})$ when $E$ is a finite graph.

math.OA↗

arXiv2019

Primitive Ideals of Labelled Graph $C^*$-algebras

Given a directed graph $E$ and a labeling $\mathcal{L}$, one forms the labelled graph $C^*$-algebra by taking a weakly left--resolving labelled space $(E, \mathcal{L}, \mathcal{B})$ and considering a universal generating family of partial isometries and projections. In this paper we provide characterization for primitive ideals of labelled graph $C^*$-algebras.

math.OA↗

arXiv2011

$C^*$-algebra of the $\mathds{Z}^n$-tree

Let $Λ= \mathbb{Z}^n$ with lexicographic ordering. $Λ$ is a totally ordered group. Let $X = Λ^+ * Λ^+$. Then $X$ is a $Λ$-tree. Analogous to the construction of graph $C^*$-algebras, we form a groupoid whose unit space is the space of ends of the tree. The $C^*$-algebra of the $Λ$-tree is defined as the $C^*$-algebra of this groupoid. We prove some properties of this $C^*$-algebra.

math.OA↗

arXiv2009

$K$-theory of $C^*$-algebras of directed graphs

For a directed graph $E$, we compute the $K$-theory of the $C^*$-algebra $C^*(E)$ from the Cuntz-Krieger generators and relations. First we compute the $K$-theory of the crossed product $C^*(E)\times_γ\IT$, and then using duality and the Pimsner-Voiculescu exact sequence we compute the $K$-theory of $C^*(E)\otimes\CK \cong (C^*(E)\times\IT)\times\IZ$. The method relies on the decomposition of $C^*(E)$ as an inductive limit of Toeplitz graph $C^*$-algebras, indexed by the finite subgraphs of $E$. The proof and result require no special asssumptions about the graph, and is given in graph-theoretic terms. This can be helpful if the graph is described by pictures rather than by a matrix.

math.OA↗

arXiv2003

Characterizing Liminal And Type I Graph C*-Algebras

We prove that the C*-algebra of a directed graph $E$ is liminal iff the graph satisfies the finiteness condition: if $p$ is an infinite path or a path ending with a sink or an infinite emitter, and if $v$ is any vertex, then there are only finitely many paths starting with $v$ and ending with a vertex in $p$. Moreover, C*(E) is Type I precisely when the circuits of $E$ are either terminal or transitory, i.e., $E$ has no vertex which is on multiple circuits, and $E$ satisfies the weaker condition: for any infinite path $λ$, there are only finitely many vertices of $λ$ that get back to $λ$ in an infinite number of ways.

math.OA↗