Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0510576

Strongly 1-bounded von Neumann algebras

Kenley Jung
View original source ↗Read source document ↗Export metadata ↓

Abstract

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $α>0$, F is $α$-bounded if the free packing $α$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x in F with finite free entropy. It is shown that if M is strongly 1-bounded, then any finite set of selfadjoint generators G for M is 1-bounded and the microstates free entropy dimension of G is less than or equal to 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and the microstates free entropy dimension is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II_1-factors which have property Gamma, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of SL_n(Z), n >2. If M and N are strongly 1-bounded and their intersection is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II_1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kenley Jung. 2005-10-31. Strongly 1-bounded von Neumann algebras. https://arxiv.org/abs/math/0510576

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Selfless Reduced Crossed Product $C^{*}$-Algebras Arising from Almost Periodic Actions

We prove that the reduced crossed product arising from an almost periodic action of an infinite countable discrete group on a unital, separable $C^*$-algebra is selfless whenever the crossed product is simple and monotracial and the group admits a topologically free extreme boundary action. We also give an alternative proof of a special case of a result of Gao, Junge, Kunnawalkam Elayavalli, Patchell, and Robert: the direct product of an exact $C^*$-simple group and a $C^*$-selfless group is $C^*$-selfless.

math.OA↗
arXiv2026

Isomorphisms and representations of noncommutative function algebras

Let $V$ be a noncommutative (NC) variety in a NC operator ball. We study the dual operator algebra $H^\infty(V)$ of bounded NC functions on $V$, and the operator algebra $A(V)$ obtained as the closure of NC polynomials in $H^\infty(V)$. We identify the space of weak-$*$ continuous finite-dimensional representations of $H^\infty(V)$ as the similarity envelope $\widetilde{V}$ of $V$, and equip $\widetilde{V}$ with a metric which turns $\widetilde{V}$ into a complete invariant for $H^\infty(V)$, in the sense that $H^\infty(V)$ is weak-$*$ continuously isomorphic to $H^\infty(W)$ if and only if $\widetilde{V}$ and $\widetilde{W}$ are NC biholomorphic via a bi-Lipschitz map. Similar results are obtained in the completely bounded category as well as for the algebras $A(V)$ when $V$ is assumed homogeneous. In the case of homogeneous varieties $V$ and $W$, we obtain a sharper classification result, by which $H^\infty(V)$ is (completely) boundedly isomorphic to $H^\infty(W)$ if and only if there is a linear map which gives a bi-Lipschitz bijection between $\widetilde{W}$ and $\widetilde{V}$. The reduction to the linear case in the generality we work in depends on showing that the existence of maps implies the existence of origin-fixing maps, which we show using a novel ``two-circle trick" of independent interest. In the important case where the varieties are the full NC operator balls we find a striking rigidity phenomenon: if $A(B_E)$ and $A(B_F)$ are completely boundedly isomorphic then $E$ and $F$ are completely isometric. We also present a variety of examples illustrating the results and their sharpness, showing how several results from the literature fail to generalize to the setting studied here.

math.OA↗
arXiv2026

Pointwise convergence of noncommutative discrete spherical ergodic averages

In this paper, we establish spherical maximal inequalities associated with commuting trace-preserving automorphisms on von Neumann algebras, extending Magyar-Stein-Wainger's discrete spherical maximal inequalities to the noncommutative setting. On a noncommutative probability space, we further prove that, under a strong ergodicity assumption, the corresponding discrete spherical averages converge bilaterally almost uniformly to $τ(x)1$ for every $x\in L_p(\mathcal M)$, $\frac{d}{d-2}<p\leq\infty$, when $d\geq5$.

math.OA↗