The rigidity problem for uniform Roe algebras
We solve the rigidity problem for uniform Roe algebras, by showing that two uniformly locally finite metric spaces with isomorphic uniform Roe algebras are bijectively coarsely equivalent.
math.OA↗
arXiv subjects
Alessandro Vignati
Publications and source records attributed to Alessandro Vignati.
The noncommutative weak Extension Principle
We introduce and study the noncommutative weak Extension Principle, a lifting principle aiming to characterise $^*$-homomorphisms between coronas of nonunital separable $\mathrm{C}^*$-algebras. While this principle fails if the Continuum Hypothesis is assumed, we show that this principle holds under mild forcing axioms such as the Open Colouring Axiom and Martin's Axiom. Further, we introduce and study the notion of nonmeagre ideals in multipliers and coronas of noncommutative $\mathrm{C}^*$-algebras, generalising the usual notion of nonmeagre ideals in $\mathcal P(\mathbb N)$.
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A note on the quasi-local algebra of expander graphs
We show that the quasi-local algebra of a coarse disjoint union of expander graphs does not contain a Cartan subalgebra isomorphic to $\ell_\infty$. N. Ozawa has recently shown that these algebras are distinct from the uniform Roe algebras of expander graphs, and our result describes a further difference.
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Rigidity of Higson coronas
We show that under mild set theoretic hypotheses we have rigidity for algebras of continuous functions over Higson coronas, topological spaces arising in coarse geometry. In particular, we show that under $\mathsf{OCA}$ and $\mathsf {MA}_{\aleph_1}$, if two uniformly locally finite metric spaces $X$ and $Y$ have homeomorphic Higson coronas $\nu X$ and $\nu Y$, then $X$ and $Y$ are coarsely equivalent, a statement which provably does not follow from $\mathsf{ZFC}$ alone.
math.LO↗
A metric lifting theorem
In a recent article by Farah and the authors, a strong lifting theorem was proved for a class of coordinate-respecting maps between reduced products of discrete structures, hereby working under mild Forcing Axioms. We generalise this lifting theorem to the metric setting.
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Generic Compacta from Relations between Finite Graphs: Theory Building and Examples
In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph structure and the relations belong to a given graph category. We relate topological properties of the spectrum to combinatorial properties of the graph categories involved. We then utilise this to exhibit elementary combinatorial constructions of well-known continua as Fra\"iss\'e limits of finite graphs in categories with relational morphisms.
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The weak Extension Principle
We prove a rigidity result for maps between \v{C}ech-Stone remainders under fairly mild forcing axioms.
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Saturation of reduced products
We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fr\'echet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not $\aleph_2$-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that $M$ is isomorphic to an ultrapower (associated with an ultrafilter on $\mathbb N$) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product $M$ is isomorphic to an ultrapower if and only if the theory of $M$ is stable. All of these conclusions apply for reduced products associated with $F_\sigma$ ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal $\mathcal Z_0$, or any other analytic P-ideal that is not $F_\sigma$, is not even $\aleph_1$-saturated if its theory is unstable.
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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.
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Trivial Isomorphisms between Reduced Products
We introduce a general method for showing under weak forcing axioms that reduced products of countable models of a theory $T$ have as few automorphisms as possible. We show that such forcing axioms imply that reduced products of countably infinite or finite fields, linear orders, trees, or random graphs have only trivial automorphisms. We also show that Todor\v{c}evi\'c's Open Colouring Axiom, $\mathsf{OCA}_{\mathrm{T}}$, implies that all automorphisms of $\mathcal{P}(\mathbb{N})/{\mathrm{Fin}}$ are trivial.
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Constructing Compacta from Posets
We develop a simple method of constructing topological spaces from countable posets with finite levels, one which applies to all second countable T_1 compacta. This results in a duality amenable to building such spaces from finite building blocks, essentially an abstract analog of classical constructions defining compacta from progressively finer open covers.
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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.
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Operator norm localization property for equi-approximable families of projections
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.
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Fraïssé theory in operator algebras
We overview the development of Fraïssé theory in the setting of continuous model theory, and some of the its recent applications to $\mathrm{C}^*$-algebra theory and functional analysis.
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Obstructions to countable saturation in corona algebras
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 $n=1$.
math.LO↗
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.
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Games on AF-algebras
We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the metric setting. We use moreover this result to build a family $\{ A_α\}_{α< ω_1}$ of pairwise non-isomorphic separable simple unital AF-algebras which satisfy $A_α\equiv_αA_β$ for every $α< β$. In particular, we obtain a set of separable simple unital AF-algebras of arbitrarily high Scott rank. Next, we give a partial converse to the aforementioned implication, showing that $A \otimes \mathcal{K} \equiv_{ω+ 2 \cdot α+2} B \otimes \mathcal{K}$ implies $K_0(A) \equiv_αK_0(B)$, for every unital $\mathrm{C}^\ast$-algebras $A$ and $B$.
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Stably projectionless Fraïssé limits
We realise the algebra $\mathcal W$, the algebra $\mathcal Z_0$ and the algebras $\mathcal Z_0\otimes A$, where $A$ is a unital UHF algebra as Fraïssé limits of suitable classes of structures. In doing so, we show that such algebras are generic objects without the use of any classification result.
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