# Operator Algebras

## New submissions

[ total of 13 entries: 1-13 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Convergence of Heisenberg Modules over Quantum 2-tori for the Modular Gromov-Hausdorff Propinquity
Comments: Third part of 1608.04881v1; second part of arXiv:1703.07073v1; split due to length. 34 pages
Subjects: Operator Algebras (math.OA)

The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a continuous family for the modular propinquity.

[2]
Title: Radial Schur multipliers on some generalisations of trees
Authors: Ignacio Vergara
Comments: 43 pages, 2 figures (created using GeoGebra)
Subjects: Operator Algebras (math.OA)

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

### Cross-lists for Tue, 20 Mar 18

[3]  arXiv:1803.06432 (cross-list from math.FA) [pdf, ps, other]
Title: Pseudo-differential operators with nonlinear quantizing functions
Subjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Operator Algebras (math.OA)

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi,$$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In particular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and $\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $\tau$ and here we investigate the corresponding calculus in the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.

[4]  arXiv:1803.06659 (cross-list from math.FA) [pdf, ps, other]
Title: New characterizations of operator monotone functions
Comments: Linear Algebra and Its Applications, 2018
Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)

If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one of these inequalities for all positive operators $A$ and $B$ and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone.
In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.

[5]  arXiv:1803.06679 (cross-list from math.GR) [pdf, ps, other]
Title: Kirchberg--Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case
Subjects: Group Theory (math.GR); Operator Algebras (math.OA)

We show that in order to prove that every second countable locally compact groups with exact reduced group C*-algebra is exact in the dynamical sense (i.e. KW-exact) it suffices to show this for totally disconnected groups.

[6]  arXiv:1803.06817 (cross-list from math.QA) [pdf, ps, other]
Title: Annular Representations of Free Product Categories
Subjects: Quantum Algebra (math.QA); Category Theory (math.CT); Operator Algebras (math.OA)

We provide a description of the annular representation category of the free product of two rigid C*-tensor categories.

[7]  arXiv:1803.06882 (cross-list from math-ph) [pdf, ps, other]
Title: The correct formulation of Gleason's theorem in quaternionic Hilbert spaces
Authors: Valter Moretti, Marco Oppio (Trento U. and TIFPA-INFN)
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Operator Algebras (math.OA); Quantum Physics (quant-ph)

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

### Replacements for Tue, 20 Mar 18

[8]  arXiv:1612.06256 (replaced) [pdf, ps, other]
Title: Triviality of Equivariant Maps in Crossed Products and Matrix Algebras
Authors: Benjamin Passer
Comments: 12 pages. Version 2 implemented the following changes: stated definitions and theorems in greater generality, cleaned up proofs, and highlighted key examples (as opposed to referring to them casually in the text)
Subjects: Operator Algebras (math.OA)
[9]  arXiv:1703.07073 (replaced) [pdf, ps, other]
Title: Heisenberg Modules over Quantum 2-tori are metrized quantum vector bundles
Comments: 38 Pages. Second part of arXiv:1608.04881v1; first part of arXiv:1703.07073v1; split due to length of paper
Subjects: Operator Algebras (math.OA)
[10]  arXiv:1706.03844 (replaced) [pdf, ps, other]
Title: Interpolation and Fatou-Zygmund property for completely Sidon subsets of discrete groups (New title: Completely Sidon sets in discrete groups)
Authors: Gilles Pisier
Comments: This new version contains a significant addition, namely the operator space version of a result of Varopoulos, showing that if the closed span of a subset of G in C*(G) is completely isomorphic to $\ell_1$ (by an arbitrary isomorphism) or if the dual operator space is exact then the set is completely Sidon. v3: more polished, longer and more detailed version with a new title
Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
[11]  arXiv:1803.04828 (replaced) [pdf, ps, other]
Title: Amenable actions of discrete quantum groups on von Neumann algebras
Subjects: Operator Algebras (math.OA)
[12]  arXiv:1602.08072 (replaced) [pdf, ps, other]
Title: Model theory of $\mathrm{C}^*$-algebras
Comments: Various bug fixes and performance improvements
Subjects: Logic (math.LO); Operator Algebras (math.OA)
[13]  arXiv:1605.07543 (replaced) [pdf, other]
Title: On tea, donuts and non-commutative geometry
Authors: Igor Nikolaev
Comments: to appear Surveys in Mathematics and its Applications
Subjects: Algebraic Geometry (math.AG); History and Overview (math.HO); Operator Algebras (math.OA)
[ total of 13 entries: 1-13 ]
[ showing up to 2000 entries per page: fewer | more ]

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