Operator Algebras

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New submissions for Tue, 14 May 24

[1]  arXiv:2405.07062 [pdf, ps, other]

Title: Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups

Authors: Robert Valente, Dilian Yang

Comments: 26 pages

Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA)

In this paper, we initiate the study of higher rank Baumslag-Solitar semigroups and their related C*-algebras. We focus on two extreme, but interesting, classes - one is related to products of odometers and the other is related to Furstenberg's $\times p,, \times q$ conjecture. For the former class, whose C*-algebras are studied by H. Li and the second author, we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting which encompasses (single-vertex) generalized Baumslag-Solitar semigroups. One of our main tools is from self-similar higher rank graphs and their C*-algebras.

[2]  arXiv:2405.07859 [pdf, other]

Title: Quasi-invariant lifts of completely positive maps for groupoid actions

Authors: Suvrajit Bhattacharjee, Marzieh Forough

Comments: Preliminary version

Subjects: Operator Algebras (math.OA)

Let $G$ be a locally compact, $\sigma$-compact, Hausdorff groupoid and $A$ be a separable, $C_0(G^{(0)})$-nuclear, $G$-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from $A$ into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for $G$-actions.

Cross-lists for Tue, 14 May 24

[3]  arXiv:2405.07888 (cross-list from math-ph) [pdf, ps, other]

Title: The fermionic massless modular Hamiltonian

Authors: Francesca La Piana, Gerardo Morsella

Comments: 22 pages, no figures

Subjects: Mathematical Physics (math-ph); Information Theory (cs.IT); Operator Algebras (math.OA)

We provide an explicit expression for the modular hamiltonian of the von Neumann algebras associated to the unit double cone for the (fermionic) quantum field theories of the 2-component Weyl (helicity 1/2) field, and of the 4-component massless Dirac and Majorana fields. To this end, we represent the one particle spaces of these theories in terms of solutions of the corresponding wave equations, and obtain the action of the modular group on them. As an application, we compute the relative entropy between the vacuum of the massless Majorana field and one particle states associated to waves with Cauchy data localized in the spatial unit ball.

[4]  arXiv:2405.07924 (cross-list from math.FA) [pdf, ps, other]

Title: Extreme points of matrix convex sets and their spanning properties

Authors: Eric Evert, Benjamin Passer, Tea Štrekelj

Comments: 26 pages

Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)

This expository article gives a survey of matrix convex sets, a natural generalization of convex sets to the noncommutative (dimension-free) setting, with a focus on their extreme points. Mirroring the classical setting, extreme points play an important role in understanding matrix convex sets, and a natural question is, "are matrix convex sets the matrix convex hull of their extreme points?" That is, does a Krein-Milman theorem hold for matrix convex sets? This question requires more care in the noncommutative setting, as there are several notions of extreme points for matrix convex sets. Three of the most important notions are matrix extreme points, matrix exposed points, and free extreme points. For each of these types of extreme points, we examine strengths and shortcomings in terms of a Krein-Milman theorem for matrix convex sets.

Replacements for Tue, 14 May 24

[5]  arXiv:2307.00435 (replaced) [pdf, ps, other]

Title: Weakly Parametric Pseudodifferential Calculus for Twisted $C^*$-dynamical Systems

Authors: Gihyun Lee, Matthias Lesch

Comments: v2: additional details in the proof of Theorem 5.2, 28 pages

Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA); Quantum Algebra (math.QA)

[6]  arXiv:2307.10108 (replaced) [pdf, ps, other]

Title: Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs

Authors: Boyu Li, Dilian Yang

Comments: 28 pages

Journal-ref: Journal of Mathematical Analysis and Applications, Volume 537, Issue 2, 15 September 2024, 128395

Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA)

[7]  arXiv:2111.01034 (replaced) [pdf, ps, other]

Title: Square-integrable representations and the coadjoint action of solvable Lie groups

Authors: Ingrid Beltita, Daniel Beltita

Comments: 29 pages; accepted for publication in the journal Forum Mathematicum

Subjects: Representation Theory (math.RT); Operator Algebras (math.OA)

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