Operator algbera seminar – University of Copenhagen

Operator algbera seminar

Speaker: Adam Dor-on (Techion)

Title: C*-envelopes of tensor algebras, dilations, Hao-Ng isomorphisms and co-universality

Abstract: An effective model, introduced by Fowler, for studying many operator algebra constructions is via product systems of C*-correspondences over quasi-lattice ordered semigroups. We use the gauge invariant uniqueness theorem of Carlsen, Larsen, Sims and Vittadello, and a new dilation technique to show that for product systems over abelian lattice-ordered semigroups the Cuntz-Nica-Pimsner algebra is always the C*-envelope of the Nica-tensor algebra. This result was implicitly conjectured in the literature, and has interesting applications in several different places. First, we resolve a problem of Skalski and Zacharias on the dilation of Nica-covariant isometric representations of product systems to unitary representations. Second, we show that the analogue of the Hao-Ng isomorphism for generalized discrete group gauge-actions in Fowler’s context has a affirmative answer in many cases, generalizing a recent result of Katsoulis in the single correspondence case. Third, we generalize C*-envelope results of Katsoulis and Kribs in the context of higher rank graphs and single correspondence case, and provide a simpler proof of a C*-envelope result of Davidson, Fuller and Kakariadis for Z^d_+ C*-dynamical systems. Our techniques show that the C*-envelope is (independently) the co-universal object sought after by Carlsen, Larsen, Sims and Vittadello, without the assumption of φ-injectivity on the product system.
(This is based on joint work with Elias Katsoulis.)