相关论文: Establishing strong 1-boundedness via non-microsta…

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $\alpha >0$, F is $\alpha$-bounded if the free packing $\alpha$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a…

Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce…

We give an general estimate for the non-microstates free entropy dimension $\delta ^{*}(X_{1},..., X_{n})$. If $X_{1},..., X_{n}$ generate a diffuse von Neumann algebra, we prove that $\delta ^{*}(X_{1},..., X_{n})\geq 1$. In the case that…

Let M be a tracial von Neumann algebra and A be a weakly dense unital C*-subalgebra of M. We say that a set X is a W*-generating set for M if the von Neumann algebra generated by X is M and that X is a C*-generating set for A if the unital…

We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying *-algebra has vanishing first L2-Betti number. The proof uses a combination of the following two key ideas to…

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 $1$-bounded entropy…

We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property…

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected…

The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung in arXiv:math/0510576 . This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann…

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there…

Suppose X is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M. If z is a selfadjoint element in M and for some selfadjoint element y in the von Neumann algebra generated by X $\delta_0(y, z) < \delta_0(y) +…

A reduction formula for compressions of von Neumann algebras arising as free products is proved. This shows that the fundamental group is all of the positive reals for some such algebras. Additionally, by taking a sort of free product with…

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free…

Let $M_n$ be a sequence of finite factors with $\dim(M_n)\rightarrow \infty$ and denote $\text{\bf M}=\Pi_\omega M_n$ their ultraproduct over a free ultrafilter $\omega$. We prove that if $\text{\bf Q}\subset \text{\bf M}$ is either an…

Let $(M, \varphi) = (M_1, \varphi_1) \ast (M_2, \varphi_2)$ be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer $M_1^{\varphi_1}$ is diffuse. We first show that any…

We show that if $\Gamma$ is an infinite finitely generated finitely presented sofic group with zero first $L^{2}$ Betti number then the von Neumann algebra $L(\Gamma)$ is strongly $1$-bounded in the sense of Jung. In particular,…

We investigate the singular subspace of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N-N subbimodule of L^{2}(M) and it contains the quasinormalizer introduced by Popa, one-sided quasinormalizer…

Suppose M is a hyperfinite von Neumann algebra with a tracial state $\phi$ and $\{a_1,...,a_n\}$ is a set of selfadjoint generators for M. We calculate $\delta_0(a_1,...,a_n)$, the modified free entropy dimension of $\{a_1,...,a_n\}$.…

In the past two decades, Sorin Popa's breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $\widetilde M \supseteq M$ by an action…

Given a suitably nested family $Z = \langle Z(m,k,\gamma) \rangle_{m,k \in \mathbb N, \gamma >0}$ of Borel subsets of matrices, and associated Borel measures and rate function, $\mu$, an entropy, $\chi^{\mu}(Z)$, is introduced which…