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Tristan Bice

Publications and source records attributed to Tristan Bice.

At least 37 records · Page 2Linked to original sources
arXiv2020

Reconstructing Etale Groupoids from Semigroups

We unify various étale groupoid reconstruction theorems such as: 1) Kumjian-Renault's reconstruction from a groupoid C*-algebra. 2) Exel's reconstruction from an ample inverse semigroup. 3) Steinberg's reconstruction from a groupoid ring. 4) Choi-Gardella-Thiel's reconstruction from a groupoid L^p-algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation < on S. The groupoid of <-ultrafilters is then isomorphic to the original groupoid G.

math.OA↗
arXiv2020

Grätzer-Hofmann-Lawson-Jung-Sünderhauf Duality

We unify several extensions of the classic Stone duality due to Grätzer, Hoffman-Lawson and Jung-Sünderhauf. Specifically we show that U-bases of locally compact sober spaces are dual to <-distributive v-predomains, where < is a transitive relation representing compact containment.

math.GN↗
arXiv2019

Distance Domains: Completeness

We explore extensions of domain theoretic concepts, replacing transitive relations with general non-symmetric distances. These lead to a generalization of Smyth completeness which we characterize in various ways analogous to our previous Yoneda completeness characterizations.

math.GN↗
arXiv2019

Nearness Posets

We extend nearness frames to posets representing bases and even subbases of $T_1$ spaces. This allows us to put a classic duality due to Wallman, between compact $T_1$ spaces and abstract simplicial complexes, into a general nearness framework. Within this framework we extend Wallman's duality to locally compact $T_1$ spaces and, with further modifications, to completely metrisable spaces. Moreover, we provide an elementary sublocale-free version of an admissibility condition due to Picado and Pultr and show how it strengthens Wallman's admissibility condition.

math.GN↗
arXiv2018

Hausdorff Tight Groupoids Generalised

We extend Exel's ample tight groupoid construction to general locally compact étale groupoids in the Hausdorff case. Moreover, we show how inverse semigroups are represented in this way as 'pseudobases' of open bisections, thus yielding a duality which encompasses various extensions of the classic Stone duality.

math.GN↗
arXiv2018

Standard homogeneous C*-algebras as compact quantum metric spaces

Given a compact metric space X and a unital C*-algebra A, we introduce a family of seminorms on the C*-algebra of continuous functions from X to A, denoted C(X, A), induced by classical Lipschitz seminorms that produce compact quantum metrics in the sense of Rieffel if and only if A is finite-dimensional. As a consequence, we are able isometrically embed X into the state space of C(X,A). Furthermore, we are able to extend convergence of compact metric spaces in the Gromov-Hausdorff distance to convergence of spaces of matrices over continuous functions on the associated compact metric spaces in Latremoliere's Gromov-Hausdorff propinquity.

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arXiv2018

General Non-Commutative Locally Compact Locally Hausdorff Stone Duality

We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to étale groupoids, obtaining a duality with an elementary class of inverse semigroups.

math.LO↗
arXiv2018

C*-Algebras With and Without $\ll$-Increasing Approximate Units

For elements $a, b$ of a C*-algebra we denote $a=ab$ by $a\ll b$. We show that all $ω_1$-unital C*-algebras have $\ll$-increasing approximate units, extending a classical result for $σ$-unital C*-algebras. We also construct (in ZFC) the first examples of C*-algebras with no $\ll$-increasing approximate unit. One of these examples is a C*-subalgebra of $\mathcal B(\ell_2(ω_1))$. These examples are, by necessity, not approximately finite dimensional (AF), but some of them are still scattered and so locally finite dimensional (LF) in the sense of Farah and Katsura. It follows that there are scattered C*-algebras which are not AF. We further show that the existence of a C*-subalgebra of $\mathcal B(\ell_2)$ with no $\ll$-increasing approximate unit or the existence of an LF but not AF C*-subalgebra of $\mathcal B(\ell_2)$ are independent of ZFC. Our examples also show that an extension of an AF algebra by an AF algebra does not have to be an AF algebra in the nonseparable case.

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arXiv2017

C*-Algebra Distance Filters

We use non-symmetric distances to give a self-contained account of C*-algebra filters and their corresponding compact projections, simultaneously simplifying and extending their general theory.

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arXiv2016

Locally Compact Stone Duality

We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices are those omitting certain types. (3) Compact clopen pseudobasic posets are characterized by separativity. We also show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations.

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arXiv2016

Yoneda Completeness

We characterize Yoneda completeness for non-symmetric distances by combinations of metric and directed completeness. One of these generalizes the Kostanek-Waszkiewicz theorem on formal balls.

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arXiv2016

*-Ring Orderings

We examine a number of *-ring orderings, generalizing classical properties of *-positive elements to *-accretives. We also examine *-rings satisfying versions of Blackadar's property (SP), generalizing some basic properties of Rickart *-rings to Blackadar *-rings.

math.RA↗
arXiv2016

A note on the Akemann-Doner and Farah-Wofsey constructions

We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable $C^*$-algebra $A$ with only separable commutative $C^*$-subalgebras. We also extend a result of Farah and Wofsey's, constructing $\aleph_1$ commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin's almost disjoint family construction.

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arXiv2014

Hereditary C*-Subalgebra Lattices

We investigate the connections between order and algebra in the hereditary C*-subalgebra lattice $\mathcal{H}(A)$ and *-annihilator ortholattice $\mathscr{P}(A)^\perp$. In particular, we characterize $\vee$-distributive elements of $\mathcal{H}(A)$ as ideals, answering a 25 year old question, allowing the quantale structure of $\mathcal{H}(A)$ to be completely determined from its lattice structure. We also show that $\mathscr{P}(A)^\perp$ is separative, allowing for C*-algebra type decompositions which are completely consistent with the original von Neumann algebra type decompositions.

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arXiv2014

*-Annihilators in Proper *-Semigroups

We generalize some basic C*-algebra and von Neumann algebra theory on hereditary C*-subalgebras and projections. In particular, we extend Murray-von Neumann equivalence from projections to *-annihilators and show that several of its important properties also extend from von Neumann algebras to proper *-semigroups. Furthermore, we show how to obtain *-annihilator decompositions of a proper *-semigroup that correspond to classical von Neumann algebra type decompositions.

math.RA↗
arXiv2014

Type Decomposition in Posets

Motivated by the classical type decomposition of von Neumann algebras, and various more recent extensions to other structures, we develop a type decomposition theory for general posets.

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arXiv2013

Annihilators and Type Decomposition in C*-Algebras

We initiate the study of annihilators in C*-algebras, showing that they are, in many ways, the best C*-algebra analogs of projections in von Neumann algebras. Using them, we obtain a type decomposition for arbitrary C*-algebras that is symmetric and completely consistent with the classical von Neumann algebra type decomposition. We also show that annihilators admit a very simple notion of equivalence that is again completely consistent with the notion of Murray-von Neumann equivalence in von Neumann algebras, sharing many of its general order theoretic properties.

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