Conference abstracts

Session S04 - Operator Algebras

July 25, 17:30 ~ 18:10

## Globalizations of partial actions and Imprimitivity Theorems

### Universidad de la República, Uruguay   -   dferraro@unorte.edu.uy

The final goal of this talk is to show how some of the well known imprimitivity theorems (as Raeburn’s Symmetric Imprimitivity Theorem) can be extended to partial actions. To that end we define proper partial actions following Buss-Echterhoff and Meyer's definitions of proper actions on C*-algebras. After that we construct, starting from a proper'' partial action $\alpha$ of $G$ on $A,$ a fixed point algebra $\mathcal{F}^\alpha(A)$ and a $\mathcal{F}^\alpha(A)-A\rtimes_{\alpha} G$ equivalence bimodule $X_\alpha.$

Under suitable assumptions, given a C*-partial action $\beta$ of $H$ on $A$ commuting with $\alpha,$ there exists a unique partial action $\widetilde{\beta}$ of $H$ on $\mathcal{F}^\alpha(A)$ cannonically induced by $\beta.$ Using F. Abadie’s notion of Morita equivalence of partial actions (as done by Curto, Muhly and Williams for global actions) we show $\mathcal{F}^\alpha(A)\rtimes_{\widehat{\beta}}H$ is Morita equivalent to $\mathcal{F}^\beta(A)\rtimes_{\widehat{\alpha}}G.$

In the seccond part of the talk we relate our imprimitivity theorems for partial actions to the problem of constructing a globalization for a given partial action on C*-algebra (or a Hilbert module). We present a necessary and sufficient condition for the existence of globalizations and, finally, we use it to investigate to what extent our imprimitivity theorems can be obtained by using Buss-Echterhoff's theorems and globalizations of partial actions.