II1 FACTORS WITH EXOTIC CENTRAL SEQUENCE ALGEBRAS

Ioana, Adrian; Spaas, Pieter

doi:10.1017/s1474748019000653

Cited by 11 publications

(4 citation statements)

References 47 publications

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“…The covering entropy Ent U (N : M) can be viewed intuitively as a measurement of the amount of tracial W * -embeddings of N into the matrix ultraproduct Q = n→U M n (C) that extend to elementary embeddings of M (compare § 4.4). This is the analog of the idea that the 1-bounded entropy h(N : M) of N in the presence of M quantifies the amount of W * -embeddings of N into Q that extend to any embedding of M. Thus, our work is motivated in part by the study of embeddings into ultraproducts, which is one theme of recent work on von Neumann algebras; for instance, see Popa [28], Goldbring [14], Ioana and Spaas [20], Atkinson and Kunnawalkam Elayavalli [2], Atkinson, Goldbring, and Kunnawalkam Elayavalli [1], Gao [11].…”

Section: Introduction 1overviewmentioning

confidence: 99%

Covering entropy for types in tracial W*-algebras

Jekel

2023

J. Log. Anal.

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We study embeddings of tracial $\mathrm{W}^*$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined \emph{$1$-bounded entropy} through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples $(X_1^{(N)},X_2^{(N)},\dots)$ having approximately the same $*$-moments as the generators $(X_1,X_2,\dots)$ of a given tracial $\mathrm{W}^*$-algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use suprema and infima, not only $*$-algebra operations and the trace | formulas such as arise in the model theory of tracial $\mathrm{W}^*$-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original $1$-bounded entropy, we show that if $\mathcal{M}$ is a separable tracial $\mathrm{W}^*$-algebra with $h(\cN:\cM) \geq 0$, then there exists an embedding of $\cM$ into a matrix ultraproduct $\cQ = \prod_{n \to \cU} M_n(\C)$ such that $h(\cN:\cQ)$ is arbitrarily close to $h(\cN:\cM)$. We deduce that if all embeddings of $\cM$ into $\cQ$ are automorphically equivalent, then $\cM$ is strongly $1$-bounded and in fact has $h(\cM) \leq 0$.

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Section: Introduction 1overviewmentioning

confidence: 99%

Covering entropy for types in tracial W*-algebras

Jekel

2023

J. Log. Anal.

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show abstract

“…The covering entropy Ent U (N : M) can be viewed intuitively as a measurement of the amount of embeddings of N into the matrix ultraproduct Q = n→U M n (C) that extend to elementary embeddings of M (compare §4.4). This is the analog of the idea that the 1-bounded entropy h(N : M) of N in the presence of M quantifies the amount of W * -embeddings of N into Q that extend to any embedding of M. Thus, our work is motivated in part by the study of embeddings into ultraproducts, which is one theme of recent work on von Neumann algebras [24,14,18,2,1,12].…”

Section: Introduction 1overviewmentioning

confidence: 99%

Covering entropy for types in tracial $\mathrm{W}^*$-algebras

Jekel¹

2022

Preprint

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We study embeddings of tracial W * -algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined 1-bounded entropy through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples (X, . . . ) having approximately the same * -moments as the generators (X1, X2, . . . ) of a given tracial W * -algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use not only *algebra operations and the trace, but also suprema and infima, such as arise in the model theory of tracial W * -algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original 1-bounded entropy, we show that if M is a separable tracial W * -algebra with h(N : M) ≥ 0, then there exists an embedding of M into a matrix ultraproduct Q = n→U Mn(C) such that h(N : Q) is arbitrarily close to h(N : Q). We deduce that if all embeddings of M into Q are automorphically equivalent, then M is strongly 1-bounded and in fact has h(M) = 0.

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“…This notion of central sequences has its origin in von Neumann algebra theory and turned out to be extremely fruitful there. In fact, the central sequence algebra was a crucial tool in several breakthrough results in von Neumann algebra theory and we refer the reader to [8] and the references therein for a more detailed discussion on this topic. In the context of this paper, a special focus was put on identifying which II 1 -factors have trivial central sequence algebra (these are II 1 -factors without the so-called property Gamma) or which have abelian central sequence algebra (those for which it is not abelian are known to be the McDuff factors).…”

Section: Introductionmentioning

confidence: 99%

Commutativity of central sequence algebras

Enders¹,

Shulman²

2021

Preprint

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The question of which separable C*-algebras have abelian central sequence algebras was raised and studied by Phillips ([16]) and Ando-Kirchberg ([2]). In this paper we give a complete answer to their question:A separable C*-algebra A has abelian central sequence algebra if and only if A satisfies Fell's condition. Moreover, we introduce a higher-dimensional analogue of Fell's condition and show that it completely characterizes subhomogeneity of central sequence algebras.In contrast, we show that any non-trivial extension by compact operators has not only non-abelian but not even residually type I central sequence algebra. In particular its central sequence algebra is not type I and not residually finitedimensional (RFD). Our techniques extensively use properties of nilpotent elements in C*-algebras.

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II₁ FACTORS WITH EXOTIC CENTRAL SEQUENCE ALGEBRAS

Cited by 11 publications

References 47 publications

Covering entropy for types in tracial W<sup>*</sup>-algebras

Covering entropy for types in tracial W<sup>*</sup>-algebras

Covering entropy for types in tracial $\mathrm{W}^*$-algebras

Commutativity of central sequence algebras

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