arXiv subjects

Benjamin Passer

Publications and source records attributed to Benjamin Passer.

18 recordsLinked to original sources

arXiv2025

Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra

We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field.

math.FA↗

arXiv2024

Extreme points of matrix convex sets and their spanning properties

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 matrix convexity, and a natural question is, ``are matrix convex sets the (closed) matrix convex hull of their extreme points?" That is, does a Krein-Milman theorem hold in this setting? This question requires some care, as there are several notions of extreme points for matrix convex sets. Three of the most prevalent 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. Of particular note is the fact that these extreme points are all finite-dimensional in nature. As such, a large amount of our discussion is about free spectrahedra, which are matrix convex sets determined by a linear matrix inequality.

math.FA↗

arXiv2024

Continuity and equivariant dimension

We study the local-triviality dimensions of actions on $C^*$-algebras, which are invariants developed for noncommutative Borsuk-Ulam theory. While finiteness of the local-triviality dimensions is known to guarantee freeness of an action, we show that free actions need not have finite weak local-triviality dimension. Moreover, the local-triviality dimensions of a continuous field may be greater than those of its individual fibers, and the dimensions may fail to vary continuously across the fibers. However, in certain circumstances upper semicontinuity of the weak local-triviality dimension is guaranteed. We examine these results and counterexamples with a focus on noncommutative tori and noncommutative spheres, both in terms of computation and theory.

math.OA↗

arXiv2021

Complex Free Spectrahedra, Absolute Extreme Points, and Dilations

Evert and Helton proved that real free spectrahedra are the matrix convex hulls of their absolute extreme points. However, this result does not extend to complex free spectrahedra, and we examine multiple ways in which the analogous result can fail. We also develop some local techniques to determine when matrix convex sets are not (duals of) free spectrahedra, as part of a continued study of minimal and maximal matrix convex sets and operator systems. These results apply to both the real and complex cases.

math.FA↗

arXiv2020

Strongly peaking representations and compressions of operator systems

We use Arveson's notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets which admit minimal presentations. A fully compressed separable operator system necessarily generates the C*-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.

math.OA↗

arXiv2019

Matrix range characterizations of operator system properties

For finite-dimensional operator systems $\mathcal{S}_{\mathsf T}$, ${\mathsf T} \in B({\mathcal H})^d$, we show that the local lifting property and $1$-exactness of $\mathcal{S}_{\mathsf T}$ may be characterized by measurements of the disparity between the matrix range $\mathcal{W}({\mathsf T})$ and the minimal/maximal matrix convex sets over its individual levels. We then examine these concepts from the point of view of free spectrahedra, direct sums of operator systems, and products of matrix convex sets.

math.FA↗

arXiv2019

Equivariant Dimensions of Graph C*-algebras

We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^*$-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph $C^*$-algebras. These include the Toeplitz algebra, Cuntz algebras, and $q$-deformed spheres.

math.OA↗

arXiv2018

Compressions of compact tuples

We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple $A$ is fully compressed if and only if it is multiplicity-free and the Shilov ideal is trivial, which occurs if and only if $A$ is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.

math.OA↗

arXiv2018

Invariants in Noncommutative Dynamics

When a compact quantum group $H$ coacts freely on unital $C^*$-algebras $A$ and $B$, the existence of equivariant maps $A \to B$ may often be ruled out due to the incompatibility of some invariant. We examine the limitations of using invariants, both concretely and abstractly, to resolve the noncommutative Borsuk-Ulam conjectures of Baum-Dabrowski-Hajac. Among our results, we find that for certain finite-dimensional $H$, there can be no well-behaved invariant which solves the Type 1 conjecture for all free coactions of $H$. This claim is in stark contrast to the case when $H$ is finite-dimensional and abelian. In the same vein, it is possible for all iterated joins of $H$ to be cleft as comodules over the Hopf algebra associated to $H$. Finally, two commonly used invariants, the local-triviality dimension and the spectral count, may both change in a $\theta$-deformation procedure.

math.OA↗

arXiv2018

Shape, Scale, and Minimality of Matrix Ranges

We study containment and uniqueness problems concerning matrix convex sets. First, to what extent is a matrix convex set determined by its first level? Our results in this direction quantify the disparity between two product operations, namely the product of the smallest matrix convex sets over $K_i \subseteq \mathbb{C}^d$, and the smallest matrix convex set over the product of $K_i$. Second, if a matrix convex set is given as the matrix range of an operator tuple $T$, when is $T$ determined uniquely? We provide counterexamples to results in the literature, showing that a compact tuple meeting a minimality condition need not be determined uniquely, even if its matrix range is a particularly friendly set. Finally, our results may be used to improve dilation scales, such as the norm bound on the dilation of (non self-adjoint) contractions to commuting normal operators, both concretely and abstractly.

math.OA↗

arXiv2017

Minimal and maximal matrix convex sets

To every convex body $K \subseteq \mathbb{R}^d$, one may associate a minimal matrix convex set $\mathcal{W}^{\textrm{min}}(K)$, and a maximal matrix convex set $\mathcal{W}^{\textrm{max}}(K)$, which have $K$ as their ground level. The main question treated in this paper is: under what conditions on a given pair of convex bodies $K,L \subseteq \mathbb{R}^d$ does $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$ hold? For a convex body $K$, we aim to find the optimal constant $\theta(K)$ such that $\mathcal{W}^{\textrm{max}}(K) \subseteq \theta(K) \cdot \mathcal{W}^{\textrm{min}}(K)$; we achieve this goal for all the $\ell^p$ unit balls, as well as for other sets. For example, if $\overline{\mathbb{B}}_{p,d}$ is the closed unit ball in $\mathbb{R}^d$ with the $\ell^p$ norm, then \[ \theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p - 1/2|}. \] This constant is sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which $\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$. In particular, we obtain that a convex body $K$ satisfies $\mathcal{W}^{\textrm{max}}(K) = \mathcal{W}^{\textrm{min}}(K)$ if and only if $K$ is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. We discuss and exploit these connections as well. For example, our results show that every $d$-tuple of self-adjoint operators of norm less than or equal to $1$, can be dilated to a commuting family of self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit constructions of these (and other) dilations.

math.OA↗

arXiv2016

Triviality of Equivariant Maps in Crossed Products and Matrix Algebras

We consider a "twisted" noncommutative join procedure for unital $C^*$-algebras which admit actions by a compact abelian group $G$ and its discrete abelian dual $\Gamma$, so that we may investigate an analogue of Baum-Dabrowski-Hajac noncommutative Borsuk-Ulam theory in the twisted setting. Namely, under what conditions is it guaranteed that an equivariant map $\phi$ from a unital $C^*$-algebra $A$ to the twisted join of $A$ and $C^*(\Gamma)$ cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same $K_0$ groups as even spheres and have an analogous Borsuk-Ulam theorem that is detected by $K_0$, despite the fact that the objects are not themselves deformations of a sphere. We find multiple sufficient conditions for twisted Borsuk-Ulam theorems to hold, one of which is the addition of another equivariance condition on $\phi$ that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps $\phi$ that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of Dabrowski-Hajac-Neshveyev) "modulo $k$."

math.OA↗

arXiv2016

Representation of free Herglotz functions

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.

math.OA↗

arXiv2016

Anticommutation in the Presentations of Theta-Deformed Spheres

We consider an analogue of the theta-deformed even spheres, modifying the relations demanded of the self-adjoint generator x in the usual presentation. In this analogue, x is given anticommutation relations with all of the other generators, as opposed to being central. Using even K-theory, we show that noncommutative Borsuk-Ulam theorems hold for these algebras.

math.OA↗

arXiv2016

Compact Group Actions on Topological and Noncommutative Joins

We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$\k{a}$browski, and Hajac: there are no equivariant morphisms $A \to A \circledast_\delta H$ or $H \to A \circledast_\delta H$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Here $A \circledast_\delta H$ denotes the equivariant noncommutative join of $A$ and $H$; this join procedure is a modification of the topological join that allows a free action of $H$ on $A$ to produce a free action of $H$ on $A \circledast_\delta H$. For the classical case $H = \mathcal{C}(G)$, $G$ a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold.

math.OA↗

arXiv2015

Free Actions on C*-algebra Suspensions and Joins by Finite Cyclic Groups

We present a proof for certain cases of the noncommutative Borsuk-Ulam conjectures proposed by Baum, D\k{a}browski, and Hajac. When a unital $C^*$-algebra $A$ admits a free action of $\mathbb{Z}/k\mathbb{Z}$, $k \geq 2$, there is no equivariant map from $A$ to the $C^*$-algebraic join of $A$ and the compact "quantum" group $C(\mathbb{Z}/k\mathbb{Z})$. This also resolves D\k{a}browski's conjecture on unreduced suspensions of $C^*$-algebras. Finally, we formulate a different type of noncommutative join than the previous authors, which leads to additional open problems for finite cyclic group actions.

math.OA↗

arXiv2015

A Noncommutative Borsuk-Ulam Theorem for Natsume-Olsen Spheres

Natsume-Olsen noncommutative spheres are C*-algebras which generalize C(S^k) when k is odd. These algebras admit natural actions by finite cyclic groups, and if one of these actions is fixed, any equivariant homomorphism between two Natsume-Olsen spheres of the same dimension induces a nontrivial map on odd K-theory. This result is an extended, noncommutative Borsuk-Ulam theorem in odd dimension, and just as in the topological case, this theorem has many (almost) equivalent formulations in terms of theta-deformed spheres of arbitrary dimension. In addition, we present theorems on graded Banach algebras, motivated by algebraic Borsuk-Ulam results of A. Taghavi.

math.QA↗

arXiv2013

An Approximate Version of the Jordan von Neumann Theorem for Finite Dimensional Real Normed Spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratically with n.

math.MG↗