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Marcel Scherer

Publications and source records attributed to Marcel Scherer.

15 recordsLinked to original sources
arXiv2026

Isomorphisms and representations of noncommutative function algebras

Let $V$ be a noncommutative (NC) variety in a NC operator ball. We study the dual operator algebra $H^\infty(V)$ of bounded NC functions on $V$, and the operator algebra $A(V)$ obtained as the closure of NC polynomials in $H^\infty(V)$. We identify the space of weak-$*$ continuous finite-dimensional representations of $H^\infty(V)$ as the similarity envelope $\widetilde{V}$ of $V$, and equip $\widetilde{V}$ with a metric which turns $\widetilde{V}$ into a complete invariant for $H^\infty(V)$, in the sense that $H^\infty(V)$ is weak-$*$ continuously isomorphic to $H^\infty(W)$ if and only if $\widetilde{V}$ and $\widetilde{W}$ are NC biholomorphic via a bi-Lipschitz map. Similar results are obtained in the completely bounded category as well as for the algebras $A(V)$ when $V$ is assumed homogeneous. In the case of homogeneous varieties $V$ and $W$, we obtain a sharper classification result, by which $H^\infty(V)$ is (completely) boundedly isomorphic to $H^\infty(W)$ if and only if there is a linear map which gives a bi-Lipschitz bijection between $\widetilde{W}$ and $\widetilde{V}$. The reduction to the linear case in the generality we work in depends on showing that the existence of maps implies the existence of origin-fixing maps, which we show using a novel ``two-circle trick" of independent interest. In the important case where the varieties are the full NC operator balls we find a striking rigidity phenomenon: if $A(B_E)$ and $A(B_F)$ are completely boundedly isomorphic then $E$ and $F$ are completely isometric. We also present a variety of examples illustrating the results and their sharpness, showing how several results from the literature fail to generalize to the setting studied here.

math.OA↗
arXiv2026

Commutative Counterexamples to Arveson's Hyperrigidity Conjecture

We construct counterexamples to Arveson's hyperrigidity conjecture for operator systems in commutative $C^*$-algebras. In each example, every irreducible representation of the $C^*$-envelope is a boundary representation, but hyperrigidity fails. The first example is the operator system of continuous affine functions on an infinite-dimensional weighted $\ell^p$ ball. In the second infinite-dimensional example, we start with commuting operators and construct a nontrivial dilation. The final example comes from a compact truncated epigraph of a strictly convex function and yields a finite-dimensional operator system.

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arXiv2026

Hyperrigidity III

In this paper, we study hyperrigidity for $C^*$-algebras. We will show that hyperrigidity can be expressed solely in terms of representations, without the need to involve general unital completely positive maps.

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arXiv2026

A Three-Dimensional Operator System without the Smith--Ward Property

Harris recently showed that a non-liftable injective representation into the Calkin algebra gives explicit four-dimensional operator systems in the Calkin algebra without the lifting property, and hence a counterexamples to the generalized Smith--Ward problem for four-dimensional operator systems. The main obstruction also appears in an earlier work by Paulsen on this problem. We isolate the relevant part of this argument and replace the four-dimensional operator system by a three-dimensional hyperrigid operator system inside a matrix amplification of \[ C_r^*(\F_2). \] The resulting Calkin subsystem is of the form span$\{1,q(D),q(K)\}$, where $D$ and $K$ are selfadjoint operators, and the identity map on this operator system has no unital completely positive lift. Equivalently, the operator $D+iK$ gives a counterexample to the Smith--Ward problem. By a result of Kavruk, the dual of this operator system fails to be exact, and hence is the first example of a three-dimensional operator system that is not exact.

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arXiv2026

On the spectral radius of operator tuples

In recent work, Shalit and Shamovich associated to every operator space structure $\mathcal{E}$ on $\mathbb{C}^d$ a spectral radius function $ρ_{\mathcal{E}}$ on $d$-tuples of operators. The main goal of this paper is to elucidate how this spectral radius depends on the operator space structure. Let $V = (\mathbb{C}^d, \|\cdot\|_V)$ be a normed space and let $\mathcal{E}$ be a quantization of $V$. We show that for a commuting operator tuple $X$, the spectral radius depends only on the underlying normed space; more precisely, \[ ρ_{\mathcal{E}}(X) = \max\{ \|λ\|_V : λ\in σ(X)\}, \] where $σ(X)$ denotes the joint spectrum of $X$. In contrast, we prove that if $\dim V \geq 3$, then $ρ_{\min(V)}(X) \neq ρ_{\max(V)}(X)$ already for some matrix tuple $X$. When $\mathcal{E}_1$ and $\mathcal{E}_2$ are selfadjoint operator spaces, we show that $ρ_{\mathcal{E}_1}(X) = ρ_{\mathcal{E}_2}(X)$ for all tuples $X$ implies $\mathcal{E}_1 = \mathcal{E}_2$. We present two proofs of this result; a key ingredient in one of them is a characterization, of independent interest, of $ρ_{\mathcal{E}}(A)$ in terms of the invertibility domain of the linear pencil associated with $A$. Finally, we prove that if two operator spaces give rise to the same spectral radius function, then the algebras of locally uniformly bounded NC functions on the corresponding NC unit balls coincide.

math.OA↗
arXiv2026

Empirical bounds for commuting dilations of free unitaries and the universal commuting dilation constant

For a tuple $T$ of Hilbert space operators, the 'commuting dilation constant' is the smallest number $c$ such that the operators of $T$ are a simultaneous compression of commuting normal operators of norm at most $c$. We present numerical experiments giving a strong indication that the commuting dilation constant of a pair of independent random $N{\times}N$ unitary matrices converges to $\sqrt2$ as $N \to \infty$ almost surely. Under the assumption that this is the case, we prove that the commuting dilation constant of an arbitrary pair of contractions is strictly smaller than $2$. Our experiments are based on a simple algorithm that we introduce for the purpose of computing dilation constants between tuples of matrices.

math.FA↗
arXiv2026

The Hyperrigidity Conjecture for Spectrahedra

We show that if K is a compact spectrahedron whose set of extreme points is closed, then the operator system of continuous affine functions on K is hyperrigid in the C*-algebra C(ex(K)).

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arXiv2026

A C*-cover lattice dichotomy

In this paper, we show that the lattice of C*-covers of a non-selfadjoint operator algebra is either one point or uncountable. We prove that there are non-selfadjoint operator algebras with a one-point lattice in two ways: as an explicit subalgebra of the C*-algebra of a universal contraction, and via a direct limit construction inspired by the work of Kirchberg and Wassermann for operator systems. We also establish that the C*-envelope need not have an immediate successor C*-cover in the lattice, and that a semi-Dirichlet non-selfadjoint operator algebra never has a one-point lattice.

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arXiv2025

The SOT-closure of the set of 2-isometries

We show that the set of $2$-isometries on an infinite-dimensional Hilbert space is not closed in the strong operator topology. In fact, we prove that its SOT-closure coincides with the set of all expansive operators.

math.FA↗
arXiv2024

The Hyperrigidity Conjecture for compact convex sets in $\mathbb{R}^2$

We prove that for every compact, convex subset $K\subset\mathbb{R}^2$ the operator system $A(K)$, consisting of all continuous affine functions on $K$, is hyperrigid in the C*-algebra $C(\mathrm{ex}(K))$. In particular, this result implies that the weak and strong operator topologies coincide on the set $$ \{ T\in\mathcal{B}(H);\ T\ \mathrm{normal}\ \mathrm{and}\ σ(T)\subset \mathrm{ex}(K) \}. $$ Our approach relies on geometric properties of $K$ and generalizes previous results by Brown.

math.FA↗
arXiv2024

Spectra of typical Hilbert space operators

Let $\mathcal{B}(H)$ be the bounded, linear operators on a separable Hilbert space equipped with the norm topology. A property is called typical if the set of operators fulfilling the property is co-meager. We show that having non-empty continuous spectrum is a typical property and that the operators with non-empty point spectrum form a nowhere dense set. In addition we show that the set of operators with empty point spectrum is dense and characterize the closure of the set of those operators for which the spectrum and the point spectrum coincide.

math.FA↗
arXiv2023

Spectra of Quotient Modules

We determine the Taylor spectra of quotient tuples of the $d$-shift on Drury-Arveson spaces with finite-dimensional coefficient spaces. We show the the Taylor spectrum can be described in terms of the approximate zero set of the annihilator ideal, and in terms of the pointwise behavior of the inner multiplier associated with the quotient tuple.

math.FA↗
arXiv2021

Essential spherical isometries

A result due to Williams, Stampfli and Fillmore shows that an essential isometry $T$ on a Hilbert space $\mathcal{H}$ is a compact perturbation of an isometry if and only if ind$(T)\le 0$. A recent result of S. Chavan yields an analogous characterization of essential spherical isometries $T=(T_1,\dots,T_n)\in\mathcal{B}(\mathcal{H})^n$ with dim($\bigcap_{i=1}^n\ker(T_i))\le$ dim$(\bigcap_{i=1}^n\ker(T_i^*))$. In the present note we show that in dimension $n>1$ the result of Chavan holds without any condition on the dimensions of the joint kernels of $T$ and $T^*$.

math.FA↗