Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1111.6183

Some explicit computations and models of free products

Madhushree Basu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note, we first work out some `bare hands' computations of the most elementary possible free products involving $\mathbb{C}^2 ~(=\mathbb{C} \oplus \mathbb{C} $) and $M_2 ~(= M_2(\mathbb{C}))$. Using these, we identify all free products $C \ast D$, where $C,D$ are of the form $A_1 \oplus A_2$ or $M_2(B)$; $A_1,A_2,B$ are finite von Neumann algebras, as is $A_1 \oplus A_2$ with the 'uniform trace' given by $tr(a_1, a_2) = 1/2 (tr(a_1) + tr(a_2))\}$ and $M_2(B)$ with the normalized trace given by $tr((b_{i,j}))=1/2(tr(b_{1,1}) + tr(b_{2,2}))$. Those results are then used to compute various possible free products involving certain finite dimensional von-Neumann algebras, the free-group von-Neumann algebras and the hyperfinite $II_1$ factor. In the process, we reprove Dykema's result `$R \ast R \cong LF_2$'.

Explore related subjects

math.OA

Keep this discovery

BibTeXRIS

Madhushree Basu. 2011-11-26. Some explicit computations and models of free products. https://arxiv.org/abs/1111.6183

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

KEEP EXPLORING

Related papers

arXiv2026

On the II$_{1}$ Factors of Fuchsian Groups

We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group factor: $L(\mathbb{F}_{2})=W^{*}(w)*W^{*}(v)$ for some Haar unitary $v\in L(\mathbb{F}_{2})$ that is freely independent from $w$. Combined with previous results, we conclude that for an arbitrary finitely generated torsion-free non-elementary discrete subgroup $\Gamma\subset PSL_{2}(\mathbb{R})$, $L(\Gamma)$ is a free group factor, settling a conjecture of de la Harpe and Voiculescu. This result was obtained using OpenAI's ChatGPT Pro 6.0.

math.OA↗
arXiv2026

On AF- and type I-ideals in certain crossed product C$^\ast$-algebras

We study locally finite-dimensional ideals in crossed products of totally disconnected spaces by free actions of the integers and in uniform Roe algebras of exact discrete groups. In the first case, we present a dynamical description of the largest locally finite-dimensional ideal, which turns out to coincide with the intersection of all maximal ideals. In the latter case, we provide a coarse geometric characterization of the locally finite-dimensional compact ideals. Moreover, we show that for crossed products of totally disconnected spaces by free actions of exact groups, the largest type I-ideal is locally finite-dimensional. In the case of uniform Roe algebras, we provide coarse geometric conditions for compact ideals guaranteeing that the ideal is type I and admits an embedding of a UHF-algebra, respectively.

math.OA↗
arXiv2026

Continuous family of compact quantum metric space structures from cocycle twisted crossed product $\textrm{C}^{\ast}$-algebras

We establish the existence of a three-parameter family of compact quantum metric space structures on cocycle twisted crossed products by discrete groups. We are mainly interested in the case where the acting group has exponential/subexponential growth. We prove that the family is jointly continuous with respect to the parameters when the acting group is exact. We obtain quantitative upper and lower bounds for the associated metric dimensions. In particular, the bounds are helpful to prove the failure of lower semicontinuity of the metric dimension with respect to the quantum Gromov-Hausdorff distance. We also prove invariance of metric dimension under zero quantum Gromov-Hausdorff distance.

math.OA↗