arXiv subjects

Matthew Wiersma

Publications and source records attributed to Matthew Wiersma.

18 recordsLinked to original sources

arXiv2025

On operator Connes-amenability of the Fourier-Stieltjes algebra

Runde and Spronk showed in 2004 that there are non-amenable groups $G$, including $\mathbb F_2$, {whose Fourier-Stieltjes algebra, $B(G)$,} is operator Connes-amenable. This result was surprising since the measure algebra $M(G)$ is Connes-amenable if and only if $G$ is amenable, which might lead one to guess that $B(G)$ should be operator Connes-amenable if and only if $G$ is amenable. This leads to the question: for which groups $G$ is $B(G)$ operator Connes-amenable? We make progress on this problem by {exhibiting} the first examples of groups {for which $B(G)$ is not operator Connes-amenable}. More specifically, we show that $B(G)$ is not operator Connes-amenable when $G$ is a non-compact locally compact group with property (T) and finite almost periodic compactification, or when $G$ is a discrete group without the factorization property.

math.FA↗

arXiv2024

Entropies and Poisson boundaries of random walks on groups with rapid decay

Let $G$ be a countable group and $\mu$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the $\mu$-random walk with respect to a weight $\omega$ on $G$ can be computed in terms of the asymptotic behavior of the spectral radius of $\mu$ in an ascending class of weighted group algebras, and we prove that for natural choices of $\omega$ and $\mu$, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the $\mu$-random walk can be realized as the Lyapunov exponent of $\mu$ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of $p$-pseudofunctions on $G$, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.

math.DS↗

arXiv2023

Exotic C*-algebras of geometric groups

We consider a new class of potentially exotic group C*-algebras $C^*_{PF_p^*}(G)$ for a locally compact group $G$, and its connection with the class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show $C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p\in (2,\infty)$, and the C*-algebras $C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of $C^*_{L^p}(G)$ and $C^*_{PF_p^*}(G)$. This greatly generalizes earlier results of Okayasu and the second author on the pairwise distinctness of $C^*_{L^p}(G)$ for $2 0$ (recall $A_π\subseteq B_π$).

math.OA↗

arXiv2023

New tensor products of C-algebras and characterization of type I C-algebras as rigidly symmetric C*-algebras

We construct several new classes of bifunctors $(A,B)\mapsto A\otimes_{\alpha} B$, where $A\otimes_\alpha B$ is a cross norm completion of $A\odot B$ for each pair of C*-algebras $A$ and $B$. For the first class of bifunctors considered $(A,B)\mapsto A\otimes_p B$ ($1\leq p\leq\infty$), $A\otimes_p B$ is a Banach algebra cross-norm completion of $A\odot B$ constructed in a fashion similar to $p$-pseudofunctions of a locally compact group. We also consider $\otimes_{p,q}$ for H\"older conjugate $p,q\in [1,\infty]$ -- a Banach $*$-algebra analogue of the tensor product $\otimes_p$. By taking enveloping C*-algebras of $A\otimes_{p,q} B$, we arrive at a third bifunctor $(A,B)\mapsto A\otimes_{\mathrm C^*_{p,q}} B$ where the resulting algebra $A\otimes_{\mathrm C^*_{p,q}} B$ is a C*-algebra. For groups belonging to a large class of non-amenable discrete groups possessing both the rapid decay and Haagerup property, we show that the tensor products $\mathrm C^*_{\mathrm r}(G_1)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G_2)$ coincide with a Brown-Guentner type C*-completion of $\mathrm \ell^1(G_1\times G_2)$ and conclude that if $2\leq p'<p\leq\infty$, then the canonical quotient map $\mathrm C^*_{\mathrm r}(G)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G)\to \mathrm C^*_{\mathrm r}(G)\otimes_{\mathrm C^*_{p',q'}}\mathrm C^*_{\mathrm r}(G)$ is not injective. A Banach $*$-algebra $A$ is \emph{rigidly symmetric} if $A\otimes_{\gamma} B$ is symmetric for every C*-algebra $B$. A theorem of Kugler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kugler's theorem by showing for C*-algebras $A$ and $B$ that $A\otimes_{\gamma}B$ is symmetric if and only if $A$ or $B$ is type I.

math.OA↗

arXiv2022

Traces on locally compact groups

We conduct a systematic study of traces on locally compact groups, in particular traces on their universal and reduced C*-algebras. We introduce the trace kernel, and examine its relation to the von Neumann kernel and to small-invariant neighbourhood (SIN) quotients. In doing so, we introduce the class of residually-$SIN$ groups, which contains both $SIN$ and maximally almost periodic groups. We examine in detail the trace kernel for connected groups. We study traces on reduced C*-algebras, giving a simple proof for compactly generated groups that existence of such a trace is equivalent to having an open normal amenable subgroup, and we display non-discrete groups admitting unique trace. We finish by examining amenable traces and the factorization property. We show for property (T) groups that amenable trace kernels coincide with von Neumann kernels. We show for totally disconnected groups that amenable trace separation implies the factorization property. We use amenable traces to give a simple proof that amenability of the group is equivalent to simultaneous nuclearity and possessing a trace of its reduced C*-algebra. As a final application of the results obtained in the paper, we address the embeddability of group C*-algebras into simple AF algebras. As a consequence, if a locally compact group is amenable and tracially separated (trace kernel is trivial), then its reduced C*-algebra is quasi-diagonal.

math.OA↗

arXiv2020

Cohomological obstructions to lifting properties for full group C$^*$-algebras

We develop a new method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C$^*$-algebras of countable groups with (relative) property (T). We derive that the full C$^*$-algebras of the groups $\mathbb Z^2\rtimes\text{SL}_2(\mathbb Z)$ and $\text{SL}_n(\mathbb Z)$, for $n\geq 3$, do not have the local lifting property (LLP). We also prove that the full C$^*$-algebras of a large class of groups $Γ$ with property (T), including those such that $\text{H}^2(Γ,\mathbb R)\not=0$ or $\text{H}^2(Γ,\mathbb ZΓ)\not=0$, do not have the lifting property (LP). More generally, we show that the same holds if $Γ$ admits a probability measure preserving action with non-vanishing second $\mathbb R$-valued cohomology. Finally, we prove that the full C$^*$-algebra of any non-finitely presented property (T) group fails the LP.

math.OA↗

arXiv2019

Quasi-Hermitian locally compact groups are amenable

A locally compact group $G$ is called Hermitian if the spectrum $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in L^1(G)$ satisfying $f=f^*$, and called quasi-Hermitian if $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in C_c(G)$ satisfying $f=f^*$. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of "spectral interpolation of triple Banach $*$-algebras" and applying it to a family ${\rm PF}_p^*(G)$ ($1\leq p\leq \infty$) of Banach $*$-algebras related to convolution operators that lie between $L^1(G)$ and $C^*_r(G)$, the reduced group C$^*$-algebra of $G$. We show that if $G$ is quasi-Hermitian, then ${\rm PF}_p^*(G)$ and $C^*_r(G)$ have the same spectral radius on Hermitian elements in $C_c(G)$ for $p\in (1,\infty)$, and then deduce that $G$ must be amenable. We also give an alternative proof to Jenkins' result that a discrete group containing a free sub-semigroup on two generators is not quasi-Hermitian. This, in particular, provides a dichotomy on discrete elementary amenable groups: either they are non quasi-Hermitian or they have subexponential growth. Finally, for a non-amenable group $G$ with either rapid decay or Kunze-Stein property, we prove the stronger statement that ${\rm PF}_p^*(G)$ is not "quasi-Hermitian relative to $C_c(G)$" unless $p=2$.

math.FA↗

arXiv2017

Kirchberg's factorization property for locally compact groups

A locally compact group $G$ has the factorization property if the map $$C^*(G)\odot C^*(G)\ni a\otimes b\mapsto λ(a)ρ(b)\in\mathcal B(L^2(G))$$ is continuous with respect to the minimal C*-norm. This paper seeks to initiate a rigorous study of this property in the case of locally compact groups which, in contrast to the discrete case, has been relatively untouched. A partial solution to the question of when the factorization property passes to continuous embeddings is given -- a question which traces back to Kirchberg's seminal work on the topic and is known to be false in general. It is also shown that every "residually amenably embeddable" group must necessarily have the factorization property and that an analogue of Kirchberg's characterization of the factorization property for discrete groups with property (T) holds for a more general class of groups.

math.OA↗

arXiv2017

Existence of tracial states on reduced group C*-algebras

Let $G$ be a locally compact group. It is not always the case that its reduced C*-algebra $C^*_r(G)$ admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when $G$ compactly generated. In particular for $G$ almost connected, or more generally when $C^*_r(G)$ is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups for which $C^*_r(G)$ does not admit a tracial state.

math.OA↗

arXiv2017

Weak containment by restrictions of induced representations

A QSIN group is a locally compact group $G$ whose group algebra $L^1(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if $G$ is a QSIN group, $H$ is a closed subgroup of $G$, and $π$ is a unitary representation of $H$, then $π$ is weakly contained in $(\mathrm{Ind}_H^Gπ)|_H$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if $G$ is a QSIN group which contains a copy of $\mathbb F_2$ as a closed subgroup, then $C^*(G)$ is not locally reflexive and $C^*_r(G)$ does not admit the local lifting property. Further applications are drawn to the "(weak) extendability" of Fourier spaces $A_π$ and Fourier-Stieltjes spaces $B_π$.

math.OA↗

arXiv2016

On exotic group C*-algebras

Let $Γ$ be a discrete group. A $C^*$-algebra $A$ is an exotic $C^*$-algebra (associated to $Γ$) if there exist proper surjective $C^*$-quotients $C^*(Γ)\to A\to C^*_r(Γ)$. In this paper, we show that a large class of exotic $C^*$-algebras have poor local properties. More precisely, we demonstrate the failure of local reflexitity, exactness, and local lifting property. Additionally, $A$ does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when $A$ is from the class of exotic $C^*$-algebras defined by Brown and Guentner. In order to achieve the main results of this paper, we prove a result which implies the factorization property for the class of discrete groups which are algebraic subgroups of locally compact amenable groups.

math.OA↗

arXiv2015

Weak* tensor products for von Neumann algebras

The category of $C^*$-algebras is blessed with many different tensor products. In contrast, virtually the only tensor product ever used in the category of von Neumann algebras is the normal spatial tensor product. We propose a definition of what a generic tensor product in this category should be. We call these weak* tensor products. For von Neumann algebras $M$ and $N$, there are, in general, many choices of weak* tensor completions of the algebraic tensor product $M\odot N$. In fact, we demonstrate that $M$ has the property that $M\odot N$ has a unique weak* tensor product completion for every von Neumann algebra $N$ if and only if $M$ is completely atomic, i.e., is a direct product of type I factors. This in particular implies that even abelian von Neumann algebras need not have this property. As an application of the theory developed throughout the paper, we construct $2^{\mathfrak c}$ nonequivalent weak* tensor product completions of $L^\infty(\mathbb R)\odot L^\infty(\mathbb R)$.

math.OA↗

arXiv2014

C-norms for tensor products of discrete group C-algebras

Let $Γ$ be a discrete group. We show that if $Γ$ is nonamenable, then the algebraic tensor products $C^*_r(Γ)\otimes C^*_r(Γ)$ and $C^*(Γ)\otimes C^*_r(Γ)$ do not admit unique $C^*$-norms. Moreover, when $Γ_1$ and $Γ_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(Γ_1)\otimes C^*_r(Γ_2)$ and $C^*(Γ_1)\otimes C_r^*(Γ_2)$ admit $2^{\aleph_0}$ $C^*$-norms. Analogues of these results continue to hold when these familiar group $C^*$-algebras are replaced by appropriate intermediate group $C^*$-algebras.

math.OA↗

arXiv2014

$L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups

Let $G$ be a locally compact group and $1\leq p<\infty$. A continuous unitary representation $π\!: G\to B(\mathcal{H})$ of $G$ is an $L^p$-representation if the matrix coefficient functions $s\mapsto \langle π(s)x,x\rangle$ lie in $L^p(G)$ for sufficiently many $x\in \mathcal{H}$. Brannan and Ruan defined the $L^p$-Fourier algebra $A_{L^p}(G)$ to be the set of matrix coefficient functions of $L^p$-representations. Similarly, the $L^p$-Fourier-Stieltjes algebra $B_{L^p}(G)$ is defined to be the weak*-closure of $A_{L^p}(G)$ in the Fourier-Stieltjes algebra $B(G)$. These are always ideals in the Fourier-Stieltjes algebra containing the Fourier algebra. In this paper we investigate how these spaces reflect properties of the underlying group and study the structural properties of these algebras. As an application of this theory, we characterize the Fourier-Stieltjes ideals of $SL(2,\mathbb R)$.

math.FA↗

arXiv2014

Exotic group C*-algebras

Let $Γ$ be a discrete group. When $Γ$ is nonamenable, the reduced and full group $C$*-algebras differ and it is generally believed that there should be many intermediate $C$*-algebras, however few examples are known. In this paper we give new constructions and compare existing constructions of intermediate group $C$*-algebras for both generic and specific groups $Γ$.

math.OA↗

arXiv2012

Optimal error estimates for corrected trapezoidal rules

Corrected trapezoidal rules are proved for $\int_a^b f(x)\,dx$ under the assumption that $f"\in L^p([a,b])$ for some $1\leq p\leq\infty$. Such quadrature rules involve the trapezoidal rule modified by the addition of a term $k[f'(a)-f'(b)]$. The coefficient $k$ in the quadrature formula is found that minimizes the error estimates. It is shown that when $f'$ is merely assumed to be continuous then the optimal rule is the trapezoidal rule itself. In this case error estimates are in terms of the Alexiewicz norm. This includes the case when $f"$ is integrable in the Henstock--Kurzweil sense or as a distribution. All error estimates are shown to be sharp for the given assumptions on $f"$. It is shown how to make these formulas exact for all cubic polynomials $f$. Composite formulas are computed for uniform partitions.

math.CA↗

arXiv2012

Simple derivation of basic quadrature formulas

Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be four times continuously differentiable for Simpson's rule. Errors are estimated in terms of the uniform norm of second or fourth derivatives of the integrand. The proof uses only integration by parts, applied to the second or fourth derivative of the integrand, multiplied by an appropriate polynomial or piecewise polynomial function. A corrected trapezoidal rule that includes the first derivative of the integrand at the endpoints of the integration interval is also proved in this manner, the coefficient in the error estimate being smaller than for the midpoint and trapezoidal rules. The proofs are suitable for presentation in a calculus or elementary numerical analysis class. Several student projects are suggested.

math.CA↗

arXiv2011

Quadrature rules with (not too many) derivatives

Quadrature formulas for $\int_a^b f(x) dx$ where derivative terms need only be evaluated at $a$ and $b$ in the composite rule are identified. Error bounds are given when $f:[a,b]\to\mathbb{R}$ satisfies $f^{(n-1)}$ is absolutely continuous so that $f^{(n)}\in L^p([a,b])$, and when $f^{(n-1)}$ is merely continuous.

math.CA↗