Using computations in the bidual of
B
(
L
2
M
)
we develop a new technique at the von Neumann algebra level to upgrade relative proper proximality to full proper proximality. This is used to ...structurally classify subalgebras of
L
Γ
where
Γ
is an infinite group that is biexact relative to a finite family of subgroups
{
Λ
i
}
i
∈
I
such that each
Λ
i
is almost malnormal in
Γ
. This generalizes the result of Ding et al. (Properly proximal von Neumann algebras, 2022.
arXiv:2204.00517
) which classifies subalgebras of von Neumann algebras of biexact groups. By developing a combination with techniques from Popa’s deformation-rigidity theory we obtain a new structural absorption theorem for free products and a generalized Kurosh type theorem in the setting of properly proximal von Neumann algebras.
In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the ...universal sofic group. This addresses a question of Pǎunescu and generalizes the Elek–Szabo uniqueness theorem for sofic approximations.
We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using ...elementary tools and well-known examples from combinatorial group theory, combined with the Baire category theorem, we obtain a plethora of results demonstrating that several phenomena in group theory are generic. In effect, we provide a new topological framework for the analysis of various well known problems in group theory. We also provide a connection between genericity in these spaces, the word problem for finitely generated groups and model-theoretic forcing. Using these connections, we investigate a natural question raised by Osin: when does a certain space of enumerated groups contain a comeager isomorphism class? We obtain a sufficient condition that allows us to answer Osin’s question in the negative for the space of all enumerated groups and the space of left orderable enumerated groups. We document several open questions in connection with these considerations.
The findings reported in this paper aim to garner the interest of both model theorists and operator algebraists alike. Using a novel blend of model theoretic and operator algebraic methods, we show ...that the family of II1 factors elementarily equivalent to the hyperfinite II1 factor R all admit embeddings into RU with factorial relative commutant. This answers a long standing question of Popa for an uncountable family of II1 factors. We introduce the notion of a generalized Jung factor: a II1 factor M for which any two embeddings of M into its ultrapower MU are equivalent by an automorphism of MU. As an application of the result above, we show that R is the unique RU-embeddable generalized Jung factor. Using the concept of building von Neumann algebras by games and the recent refutation of the Connes embedding problem, we also show that there exists a generalized Jung factor which does not embed into RU. Moreover, we find that there are uncountably many non RU-embeddable generalized Jung type II1 von Neumann algebras. We study the space of embeddings modulo automorphic equivalence of a II1 factor N into an ultrapower II1 factor MU and equip it with a natural topometric structure, yielding cardinality results for this space in certain cases. These investigations are naturally connected to the super McDuff property for II1 factors: the property that the central sequence algebra is a II1 factor. We provide new examples, classification results, and assemble the present landscape of such factors. Finally, we prove a transfer theorem for inducing factorial commutants on embeddings with several applications.
Abstract
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 (CEP) 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 CEP 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 $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.
An exotic II1 factor without property Gamma Chifan, Ionuţ; Ioana, Adrian; Kunnawalkam Elayavalli, Srivatsav
Geometric and functional analysis,
2023/10, Volume:
33, Issue:
5
Journal Article
Peer reviewed
Open access
We introduce a new iterative amalgamated free product construction of II
1
factors, and use it to construct a separable II
1
factor which does not have property Gamma and is not elementarily ...equivalent to the free group factor
L
(
F
n
)
, for any
2
≤
n
≤
∞
. This provides the first explicit example of two non-elementarily equivalent II
1
factors without property Gamma. Moreover, our construction also provides the first explicit example of a II
1
factor without property Gamma that is also not elementarily equivalent to any ultraproduct of matrix algebras. Our proofs use a blend of techniques from Voiculescu’s free entropy theory and Popa’s deformation/rigidity theory.
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