Conference abstracts

Session S04 - Operator Algebras

July 25, 18:15 ~ 18:55

## On the simplicity and $K$-theory of the $L^p$ operator algebras $\mathcal O^p(Q)$

### Ma. Eugenia Rodríguez

### Universidad de Buenos Aires, Argentina - merodrig@dm.uba.ar

For $p\in[1,\infty)$ and a row finite graph $Q$, we define a class of representations $\rho$ of the Leavitt algebra $L(Q)$ on spaces of the form $L^p(X,\mu)$, which we call the spatial representations. We prove that for fixed $p$ and $Q$ such that $L(Q)$ is simple and purely infinite, the $L^p$ operator algebra $\mathcal O^p(Q)=\overline{\rho(L(Q))}$ is the same for the all spatial representations $\rho$. When the graph $Q$ is the rose with d petals, we recover the results given by C. Phillips in 2012, in particular for $p=2$ the Cuntz algebra $\mathcal O_d$ appears.

We give conditions for the simplicity of $\mathcal O^p(Q)$ as $L^p$ operator algebra and we calculate its $K$-theory.

Joint work with Guillermo Cortiñas (Universidad de Buenos Aires, Argentina).