The speaker will introduce the history and subject matter of approximately commuting operators, stating a question from Halmos, and turn to the specific context of essentially normal operators and their classification. Preparatory material includes the introduction to our key article, and Chapter 9 of Davidson's C*-algebras by Example. Additional resources of interest include Arveson's notes on extenstions of C*-algebras.
Approximately Commuting Operators Learning Seminar
Details
The University of Haifa is hosting a learning seminar concerning approximately commuting operators. Our key text with will be the 2016 JAMS article “Distance to Normal Elements in C*-algebras of Real Rank Zero,” by Kachkovskiy and Safarov. After a few weeks of introduction to the history of the questions surrounding approximately commuting operators, we will join one another in a deep reading of the Kachkovskiy-Safarov paper. We will each read from the assigned materials leading up to our meetings, and a volunteer from among us will present on the relevant content. Beginning Thursday, June 6th, we will meet weekly from 11:00-13:00 IDT in the Amado Building Room #619 on the Technion campus. We will additionally broadcast the meetings through zoom, although any interested party is welcome to travel to Haifa and stay with the community for lunch and the afternoon Israeli Non-Commutative Analysis Seminar.
Requirements
This seminar is designed to minimize prerequisites, so that it will be accessible to as broad an audience as possible. We will assume some familiarity operator theory, specifically the Spectral Theorem and polar decomposition of an operator, as well as the continuous functional calculus associated to abelian C*-algebras.
Upcoming
Past
June 6, 2024. Lucas Hall (Haifa) Introduction and basic BDF Theory
June 13, 2024. Pavel Prudnikov (Technion) Lin's Theorem
Commuting self-adjoint matrices approximate nearly commuting self-adjoint matrices. The speaker with discuss this result and provide the convenient proof contained in this article by Friis and Rordam.
June 20, 2024. Alon Dogan (Weizmann) Voiculescu's approximately commuting unitaries lack commuting approximants.
A modulation of Halmos's question asks whether pairs of approximately commuting unitaries are close to pairs of genuinely commuting untaries. The speaker will introduce an illustrative example of Voiculescu and show that this pair is bounded away from genuinely commuting unitaries. He will include some applications to Tsirleson's problem, quantum information, and will in particular introduce the general framework for matricial stability of groups and C*-algebras. Participants should prepare by reading this short article by Exel and Loring, as well as this article by Ozawa.
June 27, 2024. Boris Bilich (Haifa) The BDF Theorem
Proof of the BDF Theorem. The speaker will present the short proof that an essentially normal operator is a compact perturbation of a normal operator if and only if it has a trivial index function, following this article by Friis and Rordam.
July 4, 2024. Lucas Hall (Haifa) Overview of Distance to Normal Operators
July 11, 2024. Pavel Prudnikov (Technion) Operator Lipschitz Functions and An Extension Theorem (The tools)
The speaker will introduce some techniques in Operator Lipschitz Functions and apply these tools to derive an Extension Theorem for normal operators which maintains control over commutators.
July 18, 2024. Lucas Hall Two Auxiliary Theorems (Prosthetics)
The speaker will prove two theorems which take a normal operator T and render unitary operators with various properties. Essentially, these operators modify T only locally and maintain commutator control in an appropriate sense.
July 25, 2024. Boris Bilich (U. Haifa) Approximation by Operators with Finite Spectrum (Surgery)
Using the unitaries from the previous talk, the speaker will cut up the spectrum of a normal operator with the aim of approximating the original operator by one with finite spectrum.
Aug 1, 2024. Adam Dor-On Proof of The Main Theorem
We tie up loose ends.
Contact
email: hallluc1 at msu dot edu
Office: Wells Hall C320
Michigan State University
Department of Mathematics
619 Red Cedar Rd.
East Lansing, MI 48924