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Interactions between reversible and irreversible Operator Algebras.

Periodic Reporting for period 1 - IRIOA (Interactions between reversible and irreversible Operator Algebras.)

Reporting period: 2019-09-05 to 2021-09-04

We proposed a systematic study of interactions between self-adjoint (reversible) and non-self-adjoint (irreversible) operator algebras. The goals of this proposal were as follows: Goal A was to advance the structure theory of semigroup C*-algebras and boundary quotient C*-algebras. Goal B was to extend work of the ER with Katsoulis to the non-abelian group-graded context and determine the C*-envelope of the irreversible tensor algebras. Goal C was to obtain new classification results for reversible and irreversible operator algebras.

he issues that were addressed regarding Goal A and Goal B had to do with the required development of new coaction quantum symmetry methods, in order to solve problems on residual finite dimensionality and co-universality of irreversible operator algebras. Another issue that has to do with Goal C was to find precise connections between classification problems of irreversible and reversible operator algebras. The results obtained for this project were original and innovative, and have impacted the field. Each project led to significant transfer of knowledge between the ER and the supervisor as well as the group in operator algebras at Copenhagen University. This MSC IF has influence on the ER's career and enabled him to obtain a permanent academic position at Haifa University in Israel.

On top of all this, the ER has significantly broadened his research network in Europe, and has collaborated with internationally recognized European experts in operator algebras. These include Soren Eilers, Xin Li, Evgenios Kakariadis and Toke Carlsen. The ER has obtained new expertise in classification of C*-algebras, which has diversified his research profile in new directions of interest to experts in Europe. The ERs organizational skills have improved after organizing a conference, and he will continue to organize meetings on the topics of the proposal in 2022. The ERs supervision skills have improved after successfully co-supervising a Masters student and taking the course on supervision of MSc and BSc students. The ERs outreach activities have become more prominent with at least one outreach lecture each year, with many more to come as the ER plans to deliver outreach talks to highschool students in the north of Israel.
During the reporting period the ER organized one learning seminar and the operator algebras seminar of 2019-2020. The latter allowed the ER to invite experts on the topic of the proposal to collaborate and to exchange knowledge with existing faculty members at Copenhagen University. The ER met regularly with his supervisor Soren Eilers, and completed two papers with him on topics of Goal C. The ER was a co-organizer of the conference titled “Automorphisms and invariants of operator algebras” in Copenhagen University. The ER has supervised one Masters’ student with Soren Eilers, and has taken the course on “Supervising BSc. and MSc. students”. The ER has given several outreach lectures related to his research, including Copenhagen’s culture night and for highschool students at a Math summer camp in Israel.

This project resulted in five deliverables. Surprisingly, the first to be completed was Goal C. The second to be completed was Goal B. Unfortunately the ER did not manage to complete Goal A, because of a combination of the pandemic, and because some results were superseded in another paper. Instead of this, the ER produced two other related papers that fit the theme of Goal A as a replacement for the superseded work. This includes one paper of the ER with Clouatre on residual finite dimensionality of semigroup operator algebras, and one single-authored paper of the ER on C*-algebras associated with random walks.

In February 2020 the ER invited Kakariadis and Katsoulis to work on WP B. In March 2020 the ER organized a focused research group Banff together with Kakariadis, Katsoulis, Laca and X. Li, in which we were able to realize a Milestone for Goal B. After this was done, with this group of authors from Banff we produced a paper realizing Goal B, resolving the majority of questions related to Goal B. The final missing piece of identifying the co-action C*-envelope with the usual C*-envelope for tensor algebras of product systems was obtained in a separate paper of Sehnem, relying on the aforementioned paper.

We arrived at the milestone for Goal C sooner than anticipated, by studying a paper of Brownlowe, Laca, Robertson and Sims. This led to a unification of invariants arising from reversible and irreversible Pimsner operator algebras of C*-correspondences, realized in a paper with Geffen and Eilers. In this paper we recovered and improved the results of the paper by the four authors mentioned above. After the end of the fellowship, we will deliver another paper related to Goal C, together with Carlsen and Eilers on shift equivalences and equivariant stable isomorphisms between Cuntz-Krieger algebras.
Here we explain some of the progress beyond the state of the art, which is realized, in three example deliverables of the project.

1) Classification of irreversible and reversible operator algebras (With Shirly Geffen and S\{o}ren Eilers, accepted to Compositio Mathematica)

"We provide a systematic approach to resolving classification problems for non-self-adjoint operator algebras using classification of C*-algebras with additional structure. We apply our techniques to operator algebras arising from C*-correspondences, and more specifically from directed graphs. Using our approach, we generalize a recent result of Brownlowe, Laca, Robertson and Sims in two ways. These results are then used to resolve completely isometric and stably-completely isometric isomorphism problems (graded and non-graded) for non-self-adjoint graph algebras of row-finite graphs."

2) C*-envelopes for operator algebras with a coaction and co-universal algebras for product systems (With Evgenios Kakariadis, Elias Katsoulis, Marcelo Laca and Xin Li, in revision stage at Advances in Mathematics)

"We introduce coactions on operator algebras and prove the existence of a smallest coaction C*-algebra with containing a coaction operator algebra equivariantly. With this new notion of a coaction C*-envelope, we resolve a problem of Carlsen, Larsen, Sims and Vitadello on the existence of a co-universal C*-algebra for product systems over right LCM semigroups. Together with its identification as a reduced version of Sehnem's strong covariance algebra, we also extend the reduced Hao-Ng isomorphism theorem from previous work with Katsoulis to non-abelian product systems."

3) Finite dimensional approximations for operator algebra representations (With Raphael Clouatre)

"We develop a finite dimensional approximation theory for representations of operator algebras. This significantly improves upon previous work of Clouatre and Ramsey, solving several questions from their paper. We show that representations of many naturally arising operator algebras admit approximations by finite dimensional representations in the point strong* operator topology. This shows that in these cases the space of finite dimensional representations of operator algebras is a pivotal tool for their classification."
Co-action C*-envelope
Dimension groups
Random walk
K-theory computations