Noncommutative Analysis

Tag: C*-algebras

Seminar talk by Dor-On: Quantum symmetries in the representation theory of operator algebras


On next Thursday the Operator Algebras and Operator Seminar will convene for a talk by Adam Dor-On.

Title: Quantum symmetries in the representation theory of operator algebras

Speaker: Adam Dor-On (University of Illinois, Urbana-Champaign)


(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Zoom link: email me.


We introduce a non-self-adjoint generalization of Quigg’s notion of coaction of a discrete group G on a C*-algebra. We call these coactions “quantum symmetries” because from the point of view of quantum groups, coactions on C*-algebras are just actions of a quantum dual group of G on the C*-algebra. We introduce and develop a compatible C*-envelope, which is the smallest C*-coaction system which contains a given operator algebra coaction system, and we call it the cosystem C*-envelope.

It turns out that the new point of view of quantum symmetries of non-self-adjoint algebras is useful for resolving problems in both C*-algebra theory and non-self-adjoint operator algebra theory. We use quantum symmetries to resolve some problems left open in work of Clouatre and Ramsey on finite dimensional approximations of representations, as well as a problem of Carlsen, Larsen, Sims and Vitadello on the existence of a co-universal C*-algebra for product systems over arbitrary right LCM semigroup embedded in groups. This latter problem was resolved for abelian lattice ordered semigroups by the speaker and Katsoulis, and we extend this to arbitrary right LCM semigroups. Consequently, we are also able to extend the Hao-Ng isomorphism theorems of the speaker with Katsoulis from abelian lattice ordered semigroups to arbitrary right LCM semigroups.

*This talk is based on two papers. One with Clouatre, and another with Kakariadis, Katsoulis, Laca and X. Li.

Topological K-theory of C*-algebras for the Working Mathematician – Lecture 2 (Definitions and core examples)

This is a write-up of the second lecture in the course given by Haim Schochet. For the first lecture and explanations, see the previous post.

I will very soon figure out how to put various references online and post links to that, too.

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Topological K-theory of C*-algebras for the Working Mathematician – Lecture 1

Claude (Haim) Schochet is spending this semester at the Technion, and he kindly agreed to give a series of lectures on K-theory. This mini-course is called “Topological K-theory of C*-algebras for the Working Mathematician”.

There will be seven lectures (they take place in Amado 814, Mondays 11:00-12:30):

  1. A crash course in C*-algebras.
  2. K-theory by axioms and core examples.
  3. K-theory strengths and limitations.
  4. Payoffs in functional analysis: elliptic operators on compact spaces, essentially normal and Toeplitz operators.
  5. Payoffs in algebraic topology: bivariant K-theory by axioms, core examples, and the UCT.
  6. Modelling of groups, groupoids, and foliations.
  7. Payoffs in geometry: Atiyah-Singer and Connes index theorems.

Since the pace will be really fast and the scope very broad, I plan to write up some of the notes I take, to help myself keep track of these lectures. When I write I will probably introduce some mistakes, and this is completely my fault. I will also probably not be able to hold myself from making some silly remarks, for which only I am responsible.

I also hope that these notes I post may help someone who has missed one or several of the talks make up and come to the next one.

The first talk took place last Monday. To be honest I wasn’t 100% on my guard since I heard such crash courses so many times, I was sure that I’ve heard it all before but very soon I was in territory which is not so familiar to me (The title “crash course” was justified!). Maybe I will make up some of the things I write, or imagine that I heard them.

(The next lectures will be on stuff that is more advances and I will take better notes, and hopefully provide a more faithful representation of the actual lecture).

I will refer in short to the following references:

1. Pedersen – C*-algebras and their automorphism groups.

2. Brown and Ozawa – C*-algebras and finite dimensional approximation.

3. Davidson – C*-algebras by example.

4. Dixmier – C*algebras

5. Blackadar – K-theory for operator algebras

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Advanced Analysis, Notes 15: C*-algebras (square root)

This post contains some make–up material for the course Advanced Analysis. It is a theorem about the positive square root of a positive element in a C*-algebra which does not appear in the text book we are using. My improvisation for this in class came out kakha–kakha, so here is the clarification.

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