arXiv subjects
Publications and source records attributed to Ilijas Farah.
Within the framework of Zermelo-Fraenkel set theory without the Axiom of Choice, we establish equivalents to the assertion "the union of a countable collection of finite sets is countable" in the context of metric spaces, probability theory, and Hilbert spaces. The categorical principle underlying these equivalences is identified.
No type II$_1$ tracial von Neumann algebra has theory that admits quantifier elimination.
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.
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.
By Glimm's dichotomy, a separable, simple $\textrm{C}^*$-algebra has continuum-many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all $\textrm{C}^*$-algebras. In 2004, Akemann and Weaver gave a negative answer to Naimark's problem, using Jensen's diamond axiom $\diamondsuit_{\aleph_1}$, a powerful diagonalization principle that implies the Continuum Hypothesis ($\mathsf{CH}$). By a result of Rosenberg, a separably represented simple $\textrm{C}^*$-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple $\textrm{C}^*$-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's $\diamondsuit_{\aleph_1}$, denoted $\diamondsuit^\mathsf{Cohen}$, that holds in the original Cohen's model for the negation of $\mathsf{CH}$. We also prove that $\diamondsuit^\mathsf{Cohen}$ suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given $\textrm{C}^*$-algebra with a prescribed action on its space of pure states.
The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker versions of the operator norm localization property (ONL) which turn out to characterize those technical geometric conditions. We use this to obtain new rigidity results for nonmetrizable coarse spaces. As an application, we provide a novel partial answer to a question of White and Willett about Cartan subalgebras of uniform Roe algebras. We also study embeddings between uniform Roe algebras.
We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.
We study the extent of countable saturation for coronas of abelian C*-algebras. In particular, we show that the corona algebra of $C_0(\bbR^n)$ is countably saturated if and only if $n=1$.
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.
Extending a result of the first author and Katsura, we prove that for every UHF algebra $A$ of infinite type, in every uncountable cardinality $κ$ there are $2^κ$ nonisomorphic approximately matricial C*-algebras with the same $K_0$ group as $A$. These algebras are group \cstar-algebras `twisted' by prescribed canonical commutation relations (CCR), and they can also be considered as nonseparable generalizations of noncommutative tori.
A uniform Roe corona is the quotient of the uniform Roe algebra of a metric space by the ideal of compact operators. Among other results, we show that it is consistent with ZFC that isomorphism between uniform Roe coronas implies coarse equivalence between the underlying spaces, for the class of uniformly locally finite metric spaces which coarsely embed into a Hilbert space. Moreover, for uniformly locally finite metric spaces with property A, it is consistent with ZFC that isomorphism between the uniform Roe coronas is equivalent to bijective coarse equivalence between some of their cofinite subsets. We also find locally finite metric spaces such that the isomorphism of their uniform Roe coronas is independent of ZFC. All set-theoretic considerations in this paper are relegated to two 'black box' principles.
We prove that there exists a nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$ such that for every countable (or separable) structure $B$ in a countable language the quotient map from the reduced product associated with the Fréchet filter onto the ultrapower has a right inverse. The proof uses the Continuum Hypothesis. We characterize the ultrafilters $\mathcal U$ with this property, and show that consistently with ZFC such ultrafilters need not exist. We also prove a similar ZFC result sufficiently strong to obtain all concrete applications of the existence of a right inverse to the quotient map. Among applications, we prove a transfer theorem, answering a question of Schafhauser and Tikuisis, motivated by the Elliott classification programme. We also show that, in the category of C*-algebras, tensoring with the C*-algebra of all continuous functions on the Cantor space preserves elementarity. We also prove that tensoring with the Jiang--Su algebra or a UHF algebra does not preserve elementarity in general.
We generalize all known results on rigidity of uniform Roe algebras to the setting of arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between uniform Roe algebras of uniformly locally finite coarse spaces whose uniform Roe algebras contain only compact ghost projections implies that the base spaces are coarsely equivalent. Moreover, if one of the spaces has property A, then the base spaces are bijectively coarsely equivalent. We also provide a characterization for the existence of an embedding onto hereditary subalgebra in terms of the underlying spaces. As an application, we partially answer a question of White and Willett about Cartan subalgebras of uniform Roe algebras.
Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. Špakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.
We show that it is relatively consistent with ZFC that there exists a hyperfinite type $\mathrm{II}_1$-factor of density character $\aleph_1$ which is not isomorphic to its opposite, does not have any outer automorphisms, and has trivial fundamental group.
A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to~$\aleph_2$. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight~$\aleph_1$ are Corson compact. We use the Gelfand--Naimark duality, and our results are stated in terms of unital abelian \cstar-algebras.
In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces $X$ and $Y$, we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of $X$ into the uniform Roe algebra of $Y$.
Every $\mathrm{C}^*$-algebra, regardless of its density character, can be embedded into the Calkin algebra in a forcing extension of the universe obtained without collapsing any cardinal.