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Mark Tomforde

Publications and source records attributed to Mark Tomforde.

At least 19 recordsLinked to original sources
arXiv2025

Naimark's Problem for graph C*-algebras and Leavitt path algebras

We describe how boundary paths in a graph can be used to construct irreducible representations of the associated graph C*-algebra and the associated Leavitt path algebra. We use this construction to establish two sets of results: First, we prove that Naimark's Problem has an affirmative answer for graph C*-algebras, we prove that the algebraic analogue of Naimark's Problem has an affirmative answer for Leavitt path algebras, and we give necessary and sufficient conditions on the graphs for the hypotheses of Naimark's Problem to be satisfied. Second, we characterize when a graph C*-algebra has a countable (i.e., finite or countably infinite) spectrum, and prove that in this case the unitary equivalence classes of irreducible representations are in one-to-one correspondence with the shift-tail equivalence classes of the boundary paths of the graph.

math.RA↗
arXiv2023

Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence

Let $E$ and $F$ be finite graphs with no sinks, and $k$ any field. We show that shift equivalence of the adjacency matrices $A_E$ and $A_F$, together with an additional compatibility condition, implies that the Leavitt path algebras $L_k(E)$ and $L_k(F)$ are graded Morita equivalent. Along the way, we build a new type of $L_k(E)$--$L_k(F)$-bimodule (a bridging bimodule), which we use to establish the graded equivalence.

math.RA↗
arXiv2022

A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner Algebras

We introduce a property of C*-correspondences, which we call Condition (S), to serve as an analogue of Condition (L) of graphs. We use Condition (S) to prove a Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner algebras and obtain sufficient conditions for simplicity of Cuntz-Pimsner algebras. We also prove that if Q is a topological quiver with no sinks and X(Q) is the associated C*-correspondence, then X(Q) satisfies Condition (S) if and only if Q satisfies Condition(L). Finally, we consider several examples to compare and contrast Condition (S) with Schweizer's nonperiodic condition.

math.OA↗
arXiv2022

A Universal Representation for Quantum Commuting Correlations

We explicitly construct an Archimedean order unit space whose state space is affinely isomorphic to the set of quantum commuting correlations. Our construction only requires fundamental techniques from the theory of order unit spaces and operator systems. Our main results are achieved by characterizing when a finite set of positive contractions in an Archimedean order unit space can be realized as a set of projections on a Hilbert space.

quant-ph↗
arXiv2022

Morita equivalence for graded rings

The classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.

math.RA↗
arXiv2022

Matricial Archimedean order unit spaces and quantum correlations

We introduce a matricial analogue of an Archimedean order unit space, which we call a $k$-AOU space. We develop the category of $k$-AOU spaces and $k$-positive maps and exhibit functors from this category to the category of operator systems and completely positive maps. We also demonstrate the existence of injective envelopes and C*-envelopes in the category of $k$-AOU spaces. Finally, we show that finite-dimensional quantum correlations can be characterized in terms of states on finite-dimensional $k$-AOU spaces. Combined with previous work, this yields a reformulation of Tsirelson's conjecture in terms of operator systems and $k$-AOU spaces.

math.OA↗
arXiv2021

Reconstruction of groupoids and C*-rigidity of dynamical systems

We show how to construct a graded locally compact Hausdorff étale groupoid from a C*-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact étale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded isotropy is abelian and torsion free, then the extended Weyl groupoid of its reduced C*-algebra is isomorphic as a graded groupoid to G. In particular, two such groupoids are isomorphic as graded groupoids if and only if there is an equivariant diagonal-preserving isomorphism of their reduced C*-algebras. We introduce graded equivalence of groupoids, and establish that two graded groupoids in which the trivially graded isotropy has torsion-free abelian interior are graded equivalent if and only if there is an equivariant diagonal-preserving Morita equivalence between their reduced C*-algebras. We use these results to establish rigidity results for a number of classes of dynamical systems, including all actions of the natural numbers by local homeomorphisms of locally compact Hausdorff spaces.

math.OA↗
arXiv2018

Naimark's problem for graph C*-algebras

Naimark's problem asks whether a C*-algebra that has only one irreducible *-representation up to unitary equivalence is isomorphic to the C*-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable C*-algebras and Type I C*-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a C*-algebra with $\aleph_1$ generators that is a counterexample to Naimark's Problem. More precisely, they showed that the statement "There exists a counterexample to Naimark's Problem that is generated by $\aleph_1$ elements." is independent of the axioms of ZFC. Whether Naimark's problem itself is independent of ZFC remains unknown. In this paper we examine Naimark's problem in the setting of graph C*-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph C*-algebras as well as for C*-algebras of graphs in which each vertex emits a countable number of edges.

math.OA↗
arXiv2016

Classification of graph algebras: A selective survey

This survey reports on current progress of programs to classify graph C*-algebras and Leavitt path algebras up to Morita equivalence using K-theory. Beginning with an overview and some history, we trace the development of the classification of simple and nonsimple graph C*-algebras and state theorems summarizing the current status of these efforts. We then discuss the much more nascent efforts to classify Leavitt path algebras, and we describe the current status of these efforts as well as outline current impediments that must be solved for this classification program to progress. In particular, we give two specific open problems that must be addressed in order to identify the correct K-theoretic invariant for classification of simple Leavitt path algebras, and we discuss the significance of various possible outcomes to these open problems.

math.OA↗
arXiv2015

K-theory for Leavitt path algebras: computation and classification

We show that the long exact sequence for K-groups of Leavitt path algebras deduced by Ara, Brustenga, and Cortinas extends to Leavitt path algebras of countable graphs with infinite emitters in the obvious way. Using this long exact sequence, we compute explicit formulas for the higher algebraic K-groups of Leavitt path algebras over certain fields, including all finite fields and all algebraically closed fields. We also examine classification of Leavitt path algebras using K-theory. It is known that the K_0-group and K_1-group do not suffice to classify purely infinite simple unital Leavitt path algebras of infinite graphs up to Morita equivalence when the underlying field is the rational numbers. We prove for these Leavitt path algebras, if the underlying field is a number field (which includes the case when the field is the rational numbers), then the pair consisting of the K_0-group and the K_6-group does suffice to classify these Leavitt path algebras up to Morita equivalence.

math.KT↗
arXiv2014

The nuclear dimension of graph C*-algebras

Consider a graph C*-algebra C*(E) with a purely infinite ideal I (possibly all of C*(E)) such that I has only finitely many ideals and C*(E)/I is approximately finite dimensional. We prove that the nuclear dimension of C*(E) is 1. If I has infinitely many ideals, then the nuclear dimension of C*(E) is either 1 or 2.

math.OA↗
arXiv2014

The prime spectrum and primitive ideal space of a graph C*-algebra

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

math.OA↗
arXiv2013

A class of C*-algebras that are prime but not primitive

We establish necessary and sufficient conditions on a (not necessarily countable) graph E for the graph C*-algebra C*(E) to be primitive. Along with a known characterization of the graphs E for which C*(E) is prime, our main result provides us with a systematic method for easily producing large classes of (necessarily nonseparable) C*-algebras that are prime but not primitive. We also compare and contrast our results with similar results for Leavitt path algebras.

math.OA↗
arXiv2013

Identifying AF-algebras that are graph C*-algebras

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C*-algebra. We prove that any separable, unital, Type I C*-algebra with finitely many ideals is isomorphic to a graph C*-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C*-algebra if and only if it is a Type I C*-algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph C*-algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to M_k for some natural number k. All of these results provide vast supporting evidence for the conjecture that an AF-algebra is isomorphic to a graph C*-algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the intrigiung question of finding K-theoretical criteria for when an extension of two graph C*-algebras is again a graph C*-algebra.

math.OA↗
arXiv2013

One-sided shift spaces over infinite alphabets

We define a notion of (one-sided) shift spaces over infinite alphabets. Unlike many previous approaches to shift spaces over countable alphabets, our shift spaces are compact Hausdorff spaces. We examine shift morphisms between these shift spaces, and identify three distinct classes that generalize the shifts of finite type. We show that when our shift spaces satisfy a property that we call "row-finite", then shift morphisms on them may be identified with sliding block codes. As applications, we show that if two (possibly infinite) directed graphs have edge shifts that are conjugate, then the groupoids of the graphs are isomorphic, and the C*-algebras of the graphs are isomorphic.

math.OA↗
arXiv2013

Ideals in Graph Algebras

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C*-algebra is isomorphic to a graph C*-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C*-algebra (respectively, a Leavitt path algebra).

math.OA↗
arXiv2013

Classification of unital simple Leavitt path algebras of infinite graphs

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple Leavitt path algebras, then L_K(E) is Morita equivalent to L_K(F) if and only if K_0^\textnormal{alg} (L_K(E)) \cong K_0^\textnormal{alg} (L_K(F)) and the graphs $E$ and $F$ have the same number of singular vertices, and moreover, in this case one may transform the graph E into the graph F using basic moves that preserve the Morita equivalence class of the associated Leavitt path algebra. We also show that when K is a field with no free quotients, the condition that E and F have the same number of singular vertices may be replaced by K_1^\textnormal{alg} (L_K(E)) \cong K_1^\textnormal{alg} (L_K(F)), and we produce examples showing this cannot be done in general. We describe how we can combine our results with a classification result of Abrams, Louly, Pardo, and Smith to get a nearly complete classification of unital simple Leavitt path algebras - the only missing part is determining whether the "sign of the determinant condition" is necessary in the finite graph case. We also consider the Cuntz splice move on a graph and its effect on the associated Leavitt path algebra.

math.RA↗