arXiv subjects

Kristin Courtney

Publications and source records attributed to Kristin Courtney.

12 recordsLinked to original sources

arXiv2025

Soft inductive limits of operator systems and a noncommutative Lazar-Lindenstrauss theorem

We establish a flexible generalization of inductive systems of operator systems, which relaxes the usual transitivity (or coherence) condition to an asymptotic version thereof and allows for systems indexed over arbitrary nets. To illustrate the utility of this generalization, we highlight how such systems arise naturally from completely positive approximations of nuclear operator systems. Going further, we utilize an argument of Ding and Peterson to show that a separable operator system is nuclear if and only if it is an inductive limit of matrix algebras, generalizing a classic Theorem of Lazar and Lindenstrauss to the setting of noncommutative Choquet theory.

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arXiv2025

Connectivity for quantum graphs via quantum adjacency operators

Connectivity is a fundamental property of quantum graphs, previously studied in the operator system model for matrix quantum graphs and via graph homomorphisms in the quantum adjacency matrix model. In this paper, we develop an algebraic characterization of connectivity for general quantum graphs within the quantum adjacency matrix framework. Our approach extends earlier results to the non-tracial setting and beyond regular quantum graphs. We utilize a quantum Perron-Frobenius theorem that provides a spectral characterization of connectivity, and we further characterize connectivity in terms of the irreducibility of the quantum adjacency matrix and the nullity of the associated graph Laplacian. These results are obtained using the KMS inner product, which unifies and generalizes existing formulations.

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arXiv2024

Nuclearity and CPC*-systems

We write arbitrary separable nuclear C*-algebras as limits of inductive systems of finite-dimensional C*-algebras with completely positive connecting maps. The characteristic feature of such CPC*-systems is that the maps become more and more orthogonality preserving. This condition makes it possible to equip the limit, a priori only an operator space, with a multiplication turning it into a C*-algebra. Our concept generalizes the NF systems of Blackadar and Kirchberg beyond the quasidiagonal case.

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arXiv2024

Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

We consider a twist $E$ over an étale groupoid $G$. When $G$ is principal, we prove that the nuclear dimension of the reduced twisted groupoid $\mathrm{C}^*$-algebra is bounded by a number depending on the dynamic asymptotic dimension of $G$ and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the $\mathrm{C}^*$-algebra of $G$. Our proof uses a reduction to the unital case where $G$ has compact unit space, via a construction of ``groupoid unitizations'' $\widetilde{G}$ and $\widetilde{E}$ of $G$ and $E$ such that $\widetilde{E}$ is a twist over $\widetilde{G}$. The construction of $\widetilde G$ is for r-discrete (hence étale) groupoids $G$ which are not necessarily principal. When $G$ is étale, the dynamic asymptotic dimension of $G$ and $\widetilde{G}$ coincide. We show that the minimal unitizations of the full and reduced twisted groupoid $\mathrm{C}^*$-algebras of the twist over $G$ are isomorphic to the twisted groupoid $\mathrm{C}^*$-algebras of the twist over $\widetilde{G}$. We apply our result about the nuclear dimension of the twisted groupoid $\mathrm{C}^*$-algebra to obtain a similar bound on the nuclear dimension of the $\mathrm{C}^*$-algebra of an étale groupoid with closed orbits and abelian stability subgroups that vary continuously.

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arXiv2023

Images of Order Zero Maps

We give sufficient conditions allowing one to build a C*-algebraic structure on a self-adjoint linear subspace of a C*-algebra in such a way that the subspace is naturally identified with the resulting C*-algebra via a completely positive order zero map. This leads to necessary and sufficient conditions to realize a self-adjoint linear subspace of a C*-algebra as the image of an order zero map from a C*-algebra and to determine whether it is completely order isomorphic to a C*-algebra via an order zero map.

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arXiv2023

The local bisection hypothesis for twisted groupoid C*-algebras

In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid $G$ being effective. One of these conditions is that $G$ satisfies the "C*-algebraic local bisection hypothesis"; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras.

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arXiv2023

Completely positive approximations and inductive systems

We consider inductive systems of C*-algebras with completely positive contractive connecting maps. We define a condition, called C*-encoding, which is sufficient for the limit of the system to be completely order isomorphic to a C*-algebra and hence guarantees a unique C*-algebra associated to the limit. When the system consists of finite-dimensional C*-algebras, this condition is also necessary and thus characterizes when the limit is completely order isomorphic to a (nuclear) C*-algebra. C*-encoding systems generalize the NF systems of Blackadar and Kirchberg and the CPC*-systems of the author and Winter. Moreover, any system of completely positive approximations of a nuclear C*-algebra gives rise to a C*-encoding system. Consequently a separable C*-algebra is nuclear if and only if it is completely order isomorphic to the limit of a C*-encoding system. This gives an inductive limit description of all separable nuclear C*-algebras equivalent to the recent construction of the author and Winter but without the additional structure of order zero maps. Without these extra structural requirements, one can easily construct examples of our systems, which we demonstrate for all amenable group C*-algebras.

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arXiv2021

Reconstruction of twisted Steinberg algebras

We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

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arXiv2021

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring $R$. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid $G$ and each locally constant $2$-cocycle $σ$ on $G$ taking values in the units $R^\times$, we study the algebra $A_R(G,σ)$ consisting of locally constant compactly supported $R$-valued functions on $G$, with convolution and involution "twisted" by $σ$. We also introduce a "discretised" analogue of a twist $Σ$ over a Hausdorff étale groupoid $G$, and we show that there is a one-to-one correspondence between locally constant $2$-cocycles on $G$ and discrete twists over $G$ admitting a continuous global section. Given a discrete twist $Σ$ arising from a locally constant $2$-cocycle $σ$ on an ample Hausdorff groupoid $G$, we construct an associated twisted Steinberg algebra $A_R(G;Σ)$, and we show that it coincides with $A_R(G,σ^{-1})$. Given any discrete field $\mathbb{F}_d$, we prove a graded uniqueness theorem for $A_{\mathbb{F}_d}(G,σ)$, and under the additional hypothesis that $G$ is effective, we prove a Cuntz--Krieger uniqueness theorem and show that simplicity of $A_{\mathbb{F}_d}(G,σ)$ is equivalent to minimality of $G$.

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arXiv2019

Free products with amalgamation over central C*-subalgebras

Let A and B be C*-algebras whose quotients are all RFD, and let C be a central C*-subalgebra in both A and B. We prove that the full amalgamated free product of A and B over C is then RFD. This generalizes Korchagin's result that amalgamated free products of commutative C*-algebras are RFD. When applied to the case of a trivial amalgam, our methods recover the result of Exel-Loring for separable C*-algebras. As corollaries to our theorem, we give sufficient conditions for amalgamated free products of maximally almost periodic (MAP) groups to have RFD C*-algebras and hence to be MAP.

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arXiv2019

The universal C*-algebra of a contraction

We say that a contractive Hilbert space operator is universal if there is a natural surjection from its generated C*-algebra to the C*-algebra generated by any other contraction. A universal contraction may be irreducible or a direct sum of (even nilpotent) matrices; we sharpen the latter fact and its proof in several ways, including von Neumann-type inequalities for noncommutative *-polynomials. We also record properties of the unique C*-algebra generated by a universal contraction, and we show that it can be used similarly to C*(F_2) in various Kirchberg-like reformulations of Connes' Embedding Problem (some known, some new). Finally we prove some analogous results for universal C*-algebras of noncommuting row contractions and universal Pythagorean C*-algebras.

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arXiv2017

Elements of $C^*$-algebras Attaining Their Norm in a Finite-Dimensional Representation

We characterize the class of RFD $C^*$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^*$-algebra is finite-dimensional, which is equivalent to the $C^*$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^*$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

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