Search

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.

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 finitedimensionality". 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 equals one when the K_1-group is torsion-free, and equals two otherwise.

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.

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., The second version incorporates various referee suggestions, including a substantial improvement to the results of Section 3.2. This should be the final version, to appear in the M\"unster Journal of Mathematics

We introduce a notion of topological property (T) for \'etale 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., Comment: 56 pages. Version 2 makes (fairly minor) corrections and clarifications

We point out some mistakes in our paper [Buss, Alcides, Echterhoff, Siegfried, Willett, Rufus, The minimal exact crossed product, Documenta Math. vol. 23 (2018), p. 2043-2077]. The most severe one is a gap in our proof that the smallest exact crossed product functor, as constructed in [Buss, Alcides, Echterhoff, Siegfried, Willett, Rufus, The minimal exact crossed product, Documenta Math. vol. 23 (2018), p. 2043-2077], is automatically Morita compatible. We were not able to close this gap so far. So, at this time, we cannot conclude that the $C^*$-group algebra $C_{\epsilon_M}^*(G)=\mathbb{C}\rtimes_{\epsilon_M} G$ corresponding to the smallest exact Morita compatible crossed-product functor $\rtimes_{\epsilon_M}$ is equal to the reduced group $C^*$-algebra $C_r^*(G)$., DOCUMENTA MATHEMATICA, Vol 26 (2021), p. 1629-1632

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., Version 4 contains various small corrections and clarifications, and adds a new subsection 7.1 to clarify the proof of the main theorem, and to what extent the ingredients for this are 'local' in nature. Version 4 is the final version, to appear in Memoirs of the EMS

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 R\o{}rdam'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., Comment: Version 2 has extra material necessary for the follow-up paper, plus some additional clarifications

We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact \'etale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, L\"uck, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C*-algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum-Connes conjecture by Baum, Guentner and Willett extends to correspondences when restricted to separable G-C*-algebras. It therefore allows a descent in KK-theory for separable systems., New version has additional material on duality, and minor corrections

Withdrawn due to a likely error with the homeomorphism at line (4). Old abstract: In the monograph 'Limit Operators and their Applications in Operator Theory', the authors define the operator spectrum of a band-dominated operator T and prove that T is Fredholm if and only if all of the operators in its operator spectrum are invertible with uniformly bounded inverses. They also ask whether the uniform boundedness condition can in fact be dispensed with. In this note we answer this question affirmatively., Withdrawn due to a likely error

We provide an expository account of Guoliang Yu's property A. The piece starts from the basic definitions, and goes on to discuss closure properties of the class of property A spaces (and groups) and the relationship of property A to coarse embeddability problems, operator theory, and amenability. It finishes with some examples of non-property A spaces. We also include an annotated bibliography covering much of the related literature.