Interactions between reversible and irreversible Operator Algebras.

ended

We propose a systematic study of interactions between self-adjoint (reversible) and non-self-adjoint (irreversible) operator algebras. We shall import Arveson's C*-envelope and Hamana's non-commutative Furstenberg boundary to the group-graded contexts, and use them to obtain new structural, dilation and classification results for various operator algebras. The goals of this proposal are as follows: Goal A is to advance the structure theory of semigroup Nica C*-algebras and Crisp-Laca boundary quotient C*-algebras, with applications in semigroup theory. This will be achieved by adapting Hamana boundary techniques used in a recent breakthrough characterization of simplicity of reduced group C*-algebras due to Kennedy and Kalantar. Goal B is to extend work of the ER with Katsoulis to the non-abelian group-graded context and determine if the C*-envelope of the irreversible tensor algebras in Fowler's context is the Cuntz-Nica-Pimsner algebra. Applications include new Laca-type dilation results, Hao-Ng isomorphisms and non-commutative Takai-type duality. Goal C is to obtain new classification results for reversible and irreversible operator algebras by a two-way flow between the theories. This will be done by uncovering the hierarchy between invariants of C*-algebras with additional structure and of non-self-adjoint operator algebras, leading to insight on an open problem of Davidson and Katsoulis in dynamics. The anticipated results in the proposal are original and innovative, and will have an impact on the field well after the end of the fellowship. Each project is chosen to optimize the transfer of knowledge between the ER and the supervisor as well as the group in operator algebras at Copenhagen University. Considering the ambitious research program, supervision, outreach, as well as other activities suggested, the MSC IF will have a lasting influence on the ER's career and will allow him to obtain a permanent academic position after the completion of the fellowship.