# $L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$

@article{Brownlowe2016L\_2mathbbZL, title={\$L\_\{2,\mathbb\{Z\}\} \otimes L\_\{2,\mathbb\{Z\}\}\$ does not embed in \$L\_\{2,\mathbb\{Z\}\}\$}, author={Nathan Brownlowe and Adam P. W. S{\o}rensen}, journal={Journal of Algebra}, year={2016} }

For a commutative ring $R$ with unit we investigate the embedding of tensor product algebras into the Leavitt algebra $L_{2,R}$. We show that the tensor product $L_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}}$ does not embed in $L_{2,\mathbb{Z}}$ (as a unital $*$-algebra). We also prove a partial non-embedding result for the more general $L_{2,R} \otimes L_{2,R}$. Our techniques rely on realising Thompson's group $V$ as a subgroup of the unitary group of $L_{2,R}$.

#### 6 Citations

Étale Groupoids and Steinberg Algebras a Concise Introduction

- Mathematics
- 2020

We give a concise introduction to (discrete) algebras arising from etale groupoids (aka Steinberg algebras) and describe their close relationship with groupoid \(C^*\)-algebras. Their connection to… Expand

TENSOR PRODUCTS OF STEINBERG ALGEBRAS

- Mathematics
- Journal of the Australian Mathematical Society
- 2019

Abstract We prove that $A_{R}(G)\otimes _{R}A_{R}(H)\cong A_{R}(G\times H)$ if $G$ and $H$ are Hausdorff ample groupoids. As part of the proof, we give a new universal property of Steinberg algebras.… Expand

Generalizations, Applications, and Current Lines of Research

- Mathematics
- 2017

We conclude the book with various observations regarding three important aspects of Leavitt path algebras. First, we describe various generalizations of, and constructions related to, Leavitt path… Expand

The Basics of Leavitt Path Algebras: Motivations, Definitions and Examples

- Mathematics
- 2017

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra L K (E) for an… Expand

Leavitt R-algebras over countable graphs embed into L2,R

- Mathematics
- 2016

Abstract For a commutative ring R with unit we show that the Leavitt path algebra L R ( E ) of a graph E embeds into L 2 , R precisely when E is countable. Before proving this result we prove a… Expand

The Cuntz splice does not preserve $*$-isomorphism of Leavitt path algebras over $\mathbb{Z}$

- Mathematics
- 2015

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraic… Expand

#### References

SHOWING 1-10 OF 16 REFERENCES

Tensor products of Leavitt path algebras

- Mathematics
- 2011

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not… Expand

Free products in R. Thompson’s group V

- Mathematics
- 2009

We investigate some product structures in R. Thompson’s group V , primarily by studying the topological dynamics associated with V ’s action on the Cantor set C. We draw attention to the class D(V,C)… Expand

Leavitt R-algebras over countable graphs embed into L2,R

- Mathematics
- 2016

Abstract For a commutative ring R with unit we show that the Leavitt path algebra L R ( E ) of a graph E embeds into L 2 , R precisely when E is countable. Before proving this result we prove a… Expand

The Cuntz splice does not preserve $*$-isomorphism of Leavitt path algebras over $\mathbb{Z}$

- Mathematics
- 2015

We show that the Leavitt path algebras $L_{2,\mathbb{Z}}$ and $L_{2-,\mathbb{Z}}$ are not isomorphic as $*$-algebras. There are two key ingredients in the proof. One is a partial algebraic… Expand

Leavitt path algebras: the first decade

- Mathematics
- 2014

The algebraic structures known as Leavitt path algebras were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda… Expand

Classification of unital simple Leavitt path algebras of infinite graphs

- Mathematics
- 2012

We prove that if E and F are graphs with a finite number of vertices and an infinite number of edges, if K is a field, and if L_K(E) and L_K(F) are simple Leavitt path algebras, then L_K(E) is Morita… Expand

LEAVITT PATH ALGEBRAS OF FINITE GELFAND–KIRILLOV DIMENSION

- Mathematics
- 2012

Groebner–Shirshov Basis and Gelfand–Kirillov dimension of the Leavitt path algebra are derived.

The Isomorphism Problem for Higman-Thompson groups

- Mathematics
- 2010

We prove that the Higman-Thompson groups $G_{n,r}^+$ and $G_{m,s}^+$ are isomorphic if and only if $m=n$ and $\mbox{gcd}(n-1,r)=\mbox{gcd}(n-1,s)$.

Leavitt path algebras with coefficients in a commutative ring

- Mathematics
- 2009

Given a directed graph E we describe a method for constructing a Leavitt path algebra $L_R(E)$ whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness… Expand

Flow invariants in the classification of Leavitt path algebras

- Mathematics
- 2008

We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations… Expand