$L^p$ version of the Cuntz-Pimsner Algebras

The broad study of bounded operators acting on $L^p$ spaces saw a renewed interest about 10 years ago when Chris Phillips introduced $L^p$ analogues of the classical Cuntz algebras. Phillips was then followed by many others (Y. Choi, G. Cortiñas, E. Gardella, M. Lupini, M. E. Rodríguez, and H. Thiel) who introduced $L^p$-analogs of other known families of C*-algebras. In this talk, I will give a short introduction to $L^p$-operator algebras presenting several examples and basic constructions such as tensor products and crossed products. Then, I will present a rough outline of Phillips' construction of the $L^p$ Cuntz algebras via the Leavitt algebras. Switching gears for a moment, I will present a couple of results I have on representations of C*-correspondences on pairs of Hilbert spaces and explain how these motivate a definition for their $L^p$-analogue: $L^p$-correspondences. Finally, I will end the talk by showing how some $L^p$ correspondences give rise to $L^p$ analogues of the Cuntz-Pimsner algebras. In particular, I will show that the $L^p$ analogues of the Cuntz algebras and, most likely, $L^p$ crossed products by the integers can be obtained by looking at an $L^p$ version of the Fock space construction for $L^p$-correspondences.