A categorical approach to Cuntz-Pimsner C*-algebras

Abstract

Using a $C^*$-algebra $A$, a Hilbert $A$-module $E$ and a $C^*$-correspondence $(E,\phi_E)$ we use the language of category theory to construct $\mathcalO_{(E,\phi_E)}(J)$, the Cuntz-Pimsner representation relative to an ideal $J$. We provide a complete classification, up to isomorphism, of the bijective representations admitting a gauge action as relative Cuntz-Pimsner representations relative to some ideal. By doing this we obtain a simple proof of the gauge invariant uniqueness theorem for the Cuntz-Pimsner algebra $\mathcalO_{(E,\phi_E)}$ over $(E,\phi_E)$.