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Christopher Schafhauser

Publications and source records attributed to Christopher Schafhauser.

At least 19 recordsLinked to original sources
arXiv2026

The Nuclear Dimension of Extensions and Graph $C^*$-Algebras

We prove that the nuclear dimension of any graph $C^*$-algebra is at most two, and we compute the exact value of the nuclear dimension for a large class of graph $C^*$-algebras. In particular, this class covers all graphs with Condition (K). Our methods --- based on Szabó's recent $KK$-uniqueness theorem --- provide a unified approach to computing the nuclear dimension of various extensions. This includes the nuclear dimension of the Toeplitz algebra (first computed by Brake--Winter) and that of the Cuntz--Toeplitz algebras (first computed by Easo et al).

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arXiv2026

Regular simple nuclear C*-algebras

We prove Z-stability for all simple, separable, pure, nuclear C*-algebras. Combining this with very recent work of Thiel completes the proof of the Toms-Winter conjecture. We do this through the framework of II_1 factorial tracially complete C*-algebras. We characterise regularity for these algebras in various terms. In particular they are regular if and only if they have real rank zero and the Murray-von Neumann equivalence classes of projections correspond to continuous affine functions from the designated traces to [0,1]. As an application of our analysis of ultrapowers of tracial completions, we show simple pure C*-algebras whose quasitraces are traces are K-stable. In the amenable setting we give a local quantisation result, a la Popa, for II_1 factorial amenable tracially complete C*-algebras, which obtains hyperfiniteness from regularity conditions. Our Z-stability results follow from this together with a theorem of Matui and Sato. An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra.

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arXiv2026

Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras

We classify the unital embeddings of a unital separable nuclear $C^*$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.

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arXiv2026

Tracially Complete C*-Algebras

We introduce a new class of operator algebras -- tracially complete C*-algebras -- as a vehicle for transferring ideas and results between C*-algebras and their tracial von Neumann algebra completions. We obtain structure and classification results for amenable tracially complete C*-algebras satisfying an appropriate version of Murray and von Neumann's property gamma for II_1 factors. In a precise sense, these results fit between Connes' celebrated theorems for injective II_1 factors and the unital classification theorem for separable simple nuclear C*-algebras. The theory also underpins arguments for the known parts of the Toms-Winter conjecture.

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arXiv2025

Negative resolution to the $C^*$-algebraic Tarski problem

We compute the $K_1$-group of ultraproducts of unital, simple $C^*$-algebras with unique trace and strict comparison. As an application, we prove that the reduced free group $C^*$-algebras $C^*_r(F_m)$ and $C^*_r(F_n)$ are elementarily equivalent (i.e., have isomorphic ultrapowers) if and only if $m = n$. This settles in the negative the $C^*$-algebraic analogue of Tarski's 1945 problem for groups.

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arXiv2024

Uniform property $Γ$ and finite dimensional tracial boundaries

We prove that a C$^*$-algebra $A$ has uniform property $Γ$ if the set of extremal tracial states, $\partial_e T(A)$, is a non-empty compact space of finite covering dimension and for each $τ\in \partial_e T(A)$, the von Neumann algebra $π_τ(A)''$ arising from the GNS representation has property $Γ$.

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arXiv2024

A topology on E-theory

For separable $C^*$-algebras $A$ and $B$, we define a topology on the set $[[A, B]]$ consisting of homotopy classes of asymptotic morphisms from $A$ to $B$. This gives an enrichment of the Connes--Higson asymptotic category over topological spaces. We show that the Hausdorffization of this category is equivalent to the shape category of Dadarlat. As an application, we obtain a topology on the $E$-theory group $E(A, B)$ with properties analogous to those of the topology on $KK(A, B)$. The Hausdorffized $E$-theory group $EL(A, B) = E(A, B) / \overline{\{0\}}$ is also introduced and studied. We obtain a continuity result for the functor $EL(\,\cdot\,,B)$, which implies a new continuity result for the functor $KL(\,\cdot\,,B)$.

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arXiv2024

KK-rigidity of simple nuclear C*-algebras

It is shown that if $A$ and $B$ are unital separable simple nuclear $\mathcal Z$-stable C$^*$-algebras and there is a unital embedding $A \rightarrow B$ which is invertible on $KK$-theory and traces, then $A \cong B$. In particular, two unital separable simple nuclear $\mathcal Z$-stable C$^*$-algebras which either have real rank zero or unique trace are isomorphic if and only if they are homotopy equivalent. It is further shown that two finite strongly self-absorbing C$^*$-algebras are isomorphic if and only if they are $KK$-equivalent in a unit-preserving way.

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arXiv2024

An asymptotic homotopy lifting property

A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $π\colon E \rightarrow B$ and for every $^*$-homomorphism $ϕ\colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $πϕ$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $ϕ$. Blackadar has shown that this property holds for all semiprojective $C^*$-algebras. We show that a version of the homotopy lifting property for asymptotic morphisms holds for separable $C^*$-algebras that are sequential inductive limits of semiprojective $C^*$-algebras. It also holds for any separable $C^*$-algebra if the quotient map $π$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions.

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arXiv2023

Finite dimensional approximations of certain amalgamated free products of groups

A group is called matricial field (MF) if it admits finite dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of MF groups by showing that given two amenable groups with a common normal subgroup, the amalgamated free product is MF.

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arXiv2021

Constant-sized robust self-tests for states and measurements of unbounded dimension

We consider correlations, $p_{n,x}$, arising from measuring a maximally entangled state using $n$ measurements with two outcomes each, constructed from $n$ projections that add up to $xI$. We show that the correlations $p_{n,x}$ robustly self-test the underlying states and measurements. To achieve this, we lift the group-theoretic Gowers-Hatami based approach for proving robust self-tests to a more natural algebraic framework. A key step is to obtain an analogue of the Gowers-Hatami theorem allowing to perturb an "approximate" representation of the relevant algebra to an exact one. For $n=4$, the correlations $p_{n,x}$ self-test the maximally entangled state of every odd dimension as well as 2-outcome projective measurements of arbitrarily high rank. The only other family of constant-sized self-tests for strategies of unbounded dimension is due to Fu (QIP 2020) who presents such self-tests for an infinite family of maximally entangled states with even local dimension. Therefore, we are the first to exhibit a constant-sized self-test for measurements of unbounded dimension as well as all maximally entangled states with odd local dimension.

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arXiv2019

Decomposing nuclear maps

We show that the strengthened version of the completely positive approximation property of Brown, Carrión, and White---where the downward maps are asymptotically order zero and the upward maps are convex combinations of order zero maps---is enjoyed by every nuclear order zero map.

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arXiv2019

Subalgebras of simple AF-algebras

It is shown that if A is a separable, exact C*-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with unique trace. Modulo the UCT, this provides an abstract characterization of C*-subalgebras of simple, unital AF-algebras. As a consequence, for a countable, discrete, amenable group G acting on a second countable, locally compact, Hausdorff space X, C_0(X) \rtimes_r G embeds into a simple, unital AF-algebra if, and only if, X admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group G, the reduced group C*-algebra C*_r(G) admits a trace-preserving embedding into the universal UHF-algebra.

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arXiv2019

Noncommutative boundaries and the ideal structure of reduced crossed products

A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a "twisted" partial C*-dynamical system that encodes much of the structure of the action. This system can often be "untwisted," for example when the algebra is commutative, or when the algebra is prime and a certain specific subgroup has vanishing Mackey obstruction. In this case, we obtain relatively simple necessary and sufficient conditions for the ideal separation property. A key idea is a notion of noncommutative boundary for a C*-dynamical system that generalizes Furstenberg's notion of topological boundary for a group.

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arXiv2018

A new proof of the Tikuisis-White-Winter Theorem

A trace on a C*-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear C*-algebras is hyperfinite, it is easy to see that traces on nuclear C*-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of C*-algebras and, in particular, using a version of the Weyl-von Neumann Theorem due to Elliott and Kucerovsky.

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arXiv2017

A synchronous game for binary constraint systems

Recently, W. Slofstra proved that the set of quantum correlations is not closed. We prove that the set of synchronous quantum correlations is not closed, which implies his result, by giving an example of a synchronous game that has a perfect quantum approximate strategy but no perfect quantum strategy. We also exhibit a graph for which the quantum independence number and the quantum approximate independence number are different. We prove new characterisations of synchronous quantum approximate correlations and synchronous quantum spatial correlations. We solve the synchronous approximation problem of Dykema and the second author, which yields a new equivalence of Connes' embedding problem in terms of synchronous correlations.

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