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Scott Atkinson

Publications and source records attributed to Scott Atkinson.

8 recordsLinked to original sources
arXiv2020

Factorial relative commutants and the generalized Jung property for II$_1$ factors

We introduce the notion of a generalized Jung factor: a II$_1$ factor $M$ for which any two embeddings of $M$ into its ultrapower $M^{\mathcal U}$ are equivalent by an automorphism of $M^{\mathcal U}$. We show that $\mathcal R$ is not the unique generalized Jung factor but is the unique $\mathcal R^{\mathcal U}$-embeddable generalized Jung factor. We use model-theoretic techniques to obtain these results. Integral to the techniques used is the result that if $M$ is elementarily equivalent to $\mathcal R$, then any elementary embedding of $M$ into $\mathcal R^{\mathcal U}$ has factorial relative commutant. This answers a long-standing question of Popa for an uncountable family of II$_1$ factors. We also provide new examples and results about the notion of super McDuffness, which is a strengthening of the McDuff property for II$_1$ factors.

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arXiv2019

On ultraproduct embeddings and amenability for tracial von Neumann algebras

We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes Embedding Problem is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the Connes Embedding Problem is amenable if and only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\text{Out}(N\otimes N)$ on $\mathbb{H}$om$(N\otimes N, \prod_{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.

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arXiv2018

Some results on tracial stability and graph products

We establish the tracial stability of a certain class of graph products of C*-algebras. This result involves the development of the "pincushion class" of finite graphs. We then apply this result in two ways. The first application yields a selective version of Lin's Theorem for almost commuting operators. The second application addresses some approximation properties of right-angled Artin groups. In particular, we show that the full C*-algebra of any right-angled Artin group is quasidiagonal and thus has a non-trivial amenable trace, and then we apply tracial stability to show when these amenable traces are in fact locally finite dimensional.

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arXiv2018

On graph products of multipliers and the Haagerup property for $C^*$-dynamical systems

We consider the notion of the graph product of actions of groups $\left\{G_v\right\}$ on a $C^*$-algebra $\mathcal{A}$ and show that under suitable commutativity conditions the graph product action $\bigstar_\Gamma \alpha_v: \bigstar_\Gamma G_v \curvearrowright \mathcal{A}$ has the Haagerup property if each action $\alpha_v: G_v \curvearrowright \mathcal{A}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive definite multipliers is positive definite. These results have impacts on left transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.

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arXiv2017

Graph products of completely positive maps

We define the graph product of unital completely positive maps on a universal graph product of unital C*-algebras and show that it is unital completely positive itself. To accomplish this, we introduce the notion of the non-commutative length of a word, and we obtain a Stinespring construction for concatenation. This result yields the following consequences. The graph product of positive-definite functions is positive-definite. A graph product version of von Neumann's Inequality holds. Graph independent contractions on a Hilbert space simultaneously dilate to graph independent unitaries.

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arXiv2016

Minimal faces and Schur's Lemma for embeddings into R^U

In the context of N. Brown's Hom(N,R^U), we establish that given \pi: N \rightarrow R^U, the dimension of the minimal face containing [\pi] is one less than the dimension of the center of the relative commutant of \pi. We also show the "convex independence" of extreme points in the sense that the convex hull of n extreme points is an n-vertex simplex. Along the way, we establish a version of Schur's Lemma for embeddings of II$_1$-factors.

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arXiv2015

Convex Sets Associated to C*-Algebras

For A a separable unital C*-algebra and M a separable McDuff II_1-factor, we show that the space Hom_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A \rightarrow M may be considered as a closed, bounded, convex subset of a separable Banach space -- a variation on N. Brown's convex structure Hom(N,R^U). When A is nuclear, Hom_w(A,M) is affinely homeomorphic to the trace space of A, but in general Hom_w(A,M) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of Hom_w(A,M) in the case where either A or M is amenable; and we give two different conditions -- one necessary and the other sufficient -- for extremality in general. The universality of C*(F_\infty) is reflected in the fact that for any unital separable A, Hom_w(A,M) may be embedded as a face in Hom_w(C*(F_\infty),M). We also extend Brown's construction to apply more generally to Hom(A,M^U).

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