arXiv subjects

Rufus Willett

Publications and source records attributed to Rufus Willett.

At least 19 recordsLinked to original sources

arXiv2026

Conditional representation stability, classification of $*$-homomorphisms, and relative eta invariants

A quasi-representation of a group is a map from the group into a matrix algebra (or similar object) that approximately satisfies the relations needed to be a representation. Work of many people starting with Kazhdan and Voiculescu, and recently advanced by Dadarlat, Eilers-Shulman-Sørensen and others, has shown that there are topological obstructions to approximating unitary quasi-representations of groups by honest representations, where `approximation' is understood to be with respect to the operator norm. The purpose of this paper is to explore whether approximation is possible if the known obstructions vanish, partially generalizing work of Gong-Lin and Eilers-Loring-Pedersen for the free abelian group of rank two, and the Klein bottle group. We show that this is possible, at least in a weak sense, for some `low-dimensional' groups including fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three manifolds. The techniques used in the paper are $K$-theoretic: they have their origin in Baum-Connes-Kasparov type assembly maps, and in the Elliott program to classify $C^*$-algebras; Kasparov's bivariant KK-theory is a crucial tool. The key new technical ingredients are: a stable uniqueness theorem in the sense of Dadarlat-Eilers and Lin that works for non-exact $C^*$-algebras; and an analysis of maps on $K$-theory with finite coefficients in terms of the relative eta invariants of Atiyah-Patodi-Singer. Despite the proofs going through $K$-theoretic machinery, the main theorems can be stated in elementary terms that do not need any $K$-theory.

math.GR↗

arXiv2026

A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps

The $K$-homology groups of a $C^*$-algebra are receptacles for information from topology, operator algebra theory, and representation theory. For applications, one often wants to know if two $K$-homology classes are the same: the simplest way to deduce this is typically via the `primary' pairing between $K$-homology and the dual theory ($K$-theory). However, this pairing will typically miss some information: for example, it cannot detect torsion elements of $K$-homology. In this paper, we introduce a `secondary' pairing between subgroups of $K$-homology and $K$-theory that takes values in $\mathbb{Q}/\mathbb{Z}$. In good cases we show that this pairing will detect all the classes in $K$-homology that are missed by the primary pairing. We then relate our secondary pairing to the relative eta invariants of Atiyah-Patodi-Singer, and to the Thomsen exact sequence and zeta maps from $C^*$-algebra classification theory.

math.KT↗

arXiv2026

The Local Lifting Property, Property FD, and stability of approximate representations

We establish Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for classes of finitely generated groups of central importance in geometric and combinatorial group theory: $3$-manifold groups, limit groups, and certain one-relator groups and right-angled Artin groups. We deduce that such groups are very flexibly stable, with respect to normalized unitarily invariant norms. In the appendix, we show that these groups also have Kechris's property (E)MD, and hence are stable in finite actions, in the selse of Gohla--Thom. The exposition is made accessible to operator algebraists and group theorists alike.

math.GR↗

arXiv2025

A trace pairing and Elliott invariant for groupoid homology

For an étale groupoid, we define a pairing between the Crainic-Moerdijk groupoid homology and the simplex of invariant Borel probability measures on the base space. The main novelty here is that the groupoid need not have totally disconnected base space, and thus the pairing can give more refined information than the measures of clopen subsets of the base space. Our principal motivation is $C^*$-algebra theory. The Elliott invariant of a $C^*$-algebra is defined in terms of $K$-theory and traces; it is fundamental in the long-running program to classify simple $C^*$-algebras (satisfying additional necessary conditions). We use our pairing to define a groupoid Elliott invariant, and show that for many interesting groupoids it agrees with the $C^*$-algebraic Elliott invariant of the groupoid $C^*$-algebra: this includes irrational rotation algebras and the $C^*$-algebras arising from orbit breaking constructions studied by the first listed author, Putnam, and Strung. These results can be thought of as establishing a refinement of Matui's HK conjecture for the relevant groupoids.

math.OA↗

arXiv2024

Controlled KK-theory, and a Milnor exact sequence

We introduce controlled $KK$-theory groups associated to a pair $(A,B)$ of separable $C^*$-algebras. Roughly, these consist of elements of the usual $K$-theory group $K_0(B)$ that approximately commute with elements of $A$. Our main results show that these groups are related to Kasparov's $KK$-groups by a Milnor exact sequence, in such a way that Rørdam's $KL$-group is identified with an inverse limit of our controlled $KK$-groups. In the case that the $C^*$-algebras involved satisfy the UCT, our Milnor exact sequence agrees with the Milnor sequence associated to a $KK$-filtration in the sense of Schochet, although our results are independent of the UCT. Applications to the UCT will be pursued in subsequent work.

math.OA↗

arXiv2024

The rational HK-conjecture: transformation groupoids and a revised version

We prove the rational HK-conjecture for a large class of transformation groupoids in the case when the relevant action has torsion-free stabilizers. A revised version of the rational HK-conjecture in the case of (possibly) torsion stabilizers is introduced and proved for a large class of transformation groupoids. In particular, this revised version holds for Scarparo's counterexamples to the original rational HK-conjecture. The key tools used are the Baum-Connes conjecture and a Chern character defined by Raven.

math.KT↗

arXiv2023

Coarse equivalence versus bijective coarse equivalence of expander graphs

We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras of coarse disjoint unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent.

math.MG↗

arXiv2023

The UCT for $C^*$-algebras with finite complexity

A $C^*$-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's $KK$-theory to a commutative $C^*$-algebra. This paper is motivated by the problem of establishing the range of validity of the UCT, and in particular, whether the UCT holds for all nuclear $C^*$-algebras. We introduce the idea of a $C^*$-algebra that "decomposes" over a class $\mathcal{C}$ of $C^*$-algebras. Roughly, this means that locally, there are approximately central elements that approximately cut the $C^*$-algebra into two $C^*$-subalgebras from $\mathcal{C}$ that have well-behaved intersection. We show that if a $C^*$-algebra decomposes over the class of nuclear, UCT $C^*$-algebras, then it satisfies the UCT. The argument is based on controlled $KK$-theory, as introduced by the authors in earlier work. Nuclearity is used via Kasparov's Hilbert module version of Voiculescu's theorem, and Haagerup's theorem that nuclear $C^*$-algebras are amenable We say that a $C^*$-algebra has finite complexity if it is in the smallest class of $C^*$-algebras containing the finite-dimensional $C^*$-algebras, and closed under decomposability; our main result implies that all $C^*$-algebras in this class satisfy the UCT. The class of $C^*$-algebras with finite complexity is large, and comes with an ordinal-number invariant measuring the complexity level. We conjecture that a $C^*$-algebra of finite nuclear dimension and real rank zero has finite complexity; this (and several other related conjectures) would imply the UCT for all separable nuclear $C^*$-algebras. We also give new local formulations of the UCT, and some other necessary and sufficient conditions for the UCT to hold for all nuclear $C^*$-algebras.

math.OA↗

arXiv2023

Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these $\mathrm{C}^*$-algebras contain embedded copies of $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for any \emph{bounded} countable (possibly finite) collection $(n_k)_k$ of natural numbers; we aim to show that they cannot contain any other von Neumann algebras. One of our main results shows that $L_\infty[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal $*$-homomorphism. In particular, it follows from the structure theory of von Neumann algebras that any von Neumann algebra which embeds into such algebra must be of the form $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for some countable (possibly finite) collection $(n_k)_k$ of natural numbers. Under additional assumptions, we also show that the sequence $(n_k)_k$ has to be bounded: in other words, the only embedded von Neumann algebras are the ``obvious'' ones.

math.OA↗

arXiv2022

Dynamic asymptotic dimension and Matui's HK conjecture

We prove that the homology groups of a principal ample groupoid vanish in dimensions greater than the dynamic asymptotic dimension of the groupoid (as a side-effect of our methods, we also give a new model of groupoid homology in terms of the Tor groups of homological algebra, which might be of independent interest). As a consequence, the K-theory of the $C^*$-algebras associated with groupoids of finite dynamic asymptotic dimension can be computed from the homology of the underlying groupoid. In particular, principal ample groupoids with dynamic asymptotic dimension at most two and finitely generated second homology satisfy Matui's HK-conjecture. We also construct explicit maps from the groupoid homology groups to the K-theory groups of their $C^*$-algebras in degrees zero and one, and investigate their properties.

math.OA↗

arXiv2022

Dynamical complexity and controlled operator K-theory

In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the $K$-theory of the associated crossed product $C^*$-algebra by splitting it up into simpler pieces and using the methods of controlled $K$-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator $K$-theory. In particular, we do not assume prior knowledge of controlled $K$-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant $K$-theory to set up.

math.KT↗

arXiv2022

$C^*$-algebras with finite complexity

Complexity rank for $C^*$-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most $n$ if you can repeatedly cut the $C^*$-algebra in half at most $n$ times, and end up with something finite dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as `two-colored local finite-dimensionality'. We first show that for separable, unital, and simple $C^*$-algebras, weak complexity rank one is equivalent to the conjunction of nuclear dimension one and real rank zero. In particular, this shows that the UCT for all nuclear $C^*$-algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero $K$-theory groups. However, we also show using a $K$-theoretic obstruction (torsion in $K_1$) that weak complexity rank one and complexity rank one are not the same in general. We then use the Kirchberg-Phillips classification theorem to compute the complexity rank of all UCT Kirchberg algebras: it is always one or two, with the rank one case occurring if and only if the $K_1$-group is torsion free.

math.OA↗

arXiv2022

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\cstu(X)$ and $\cstu(Y)$, are isomorphic as \cstar-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, we obtain that if $Γ$ and $Λ$ are finitely generated groups, then the crossed products $\ell_\infty(Γ)\rtimes_rΓ$ and $ \ell_\infty(Λ)\rtimes_rΛ$ are isomorphic if and only if $Γ$ and $Λ$ are bi-Lipschitz equivalent.

math.OA↗

arXiv2022

Amenability and weak containment for actions of locally compact groups on $C^*$-algebras

In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long standing open problem about the equivalence of topological amenability and measurewise amenability for a second countable $G$-space $X$. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes a theorem of Matsumura: we show that for an action of an exact locally compact group $G$ on a locally compact space $X$ the full and reduced crossed products $C_0(X)\rtimes_\max G$ and $C_0(X)\rtimes_{\operatorname{red}} G$ coincide if and only if the action of $G$ on $X$ is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative $C^*$-algebras. Finally, we study amenability as it relates to more detailed structure in the case of $C^*$-algebras that fibre over an appropriate $G$-space $X$, and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L)LP, and the equivariant versions of injectivity and the WEP.

math.OA↗

arXiv2021

The UCT problem for nuclear $C^\ast$-algebras

In recent years, a large class of nuclear $C^\ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

math.OA↗

arXiv2021

Topological property (T) for groupoids

We introduce a notion of topological property (T) for étale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the existence of so-called Kazhdan projections in both maximal and reduced groupoid $C^*$-algebras, and explore applications of this to exactness, K-exactness, and the Baum-Connes conjecture. We also study various examples, and discuss the relationship with other notions of property (T) for groupoids and with a-T-menability.

math.OA↗

arXiv2020

Approximate ideal structures and K-theory

We introduce a notion of approximate ideal structure for a $C^*$-algebra, and use it as a tool to study $K$-theory groups. The notion is motivated by the classical Mayer-Vietoris sequence, by the theory of nuclear dimension as introduced by Winter and Zacharias, and by the theory of dynamical complexity introduced by Guentner, Yu, and the author. A major inspiration for our methods comes from recent work of Oyono-Oyono and Yu in the setting of controlled $K$-theory of filtered C*-algebras; we do not, however, use that language in this paper. We give two main applications. The first is a vanishing result for $K$-theory that is relevant to the Baum-Connes conjecture. The second is a permanence result for the Künneth formula in $C^*$-algebra $K$-theory: roughly, this says that if $A$ can be decomposed into a pair of subalgebras $(C,D)$ such that $C$, $D$, and $C\cap D$ all satisfy the Künneth formula, then $A$ itself satisfies the Künneth formula.

math.OA↗

arXiv2020

The minimal exact crossed product

Given a locally compact group $G$, we study the smallest exact crossed-product functor $(A,G,α)\mapsto A\rtimes_{\mathcal E} G$ on the category of $G$-$C^*$-dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor $\rtimes_{\mathcal{E}}$ as introduced by Baum, Guentner, and Willett in their reformulation of the Baum-Connes conjecture (see [2]). We show that the corresponding group algebra $C_{\mathcal{E}}^*(G)$ always coincides with the reduced group algebra, thus showing that the new formulation of the Baum-Connes conjecture coincides with the classical one in the case of trivial coefficients. Erratum: After publication of this manuscript, some gaps have unfortunately been found affecting some parts of the paper. We therefore included an appendix with an erratum at the end of this paper explaining the mistakes and keeping the original published version unchanged.

math.OA↗