Mini-workshop: Commutative subalgebras, ideals and actions, July 23-24, 2012, Lund

# General info

The main topic of this mini-workshop will be the role played by (maximal) commutative subalgebras in various types of noncommutative rings and (operator) algebras. We will also focus on connections to properties of actions and the ideal structure of the algebras involved. This event will take place at the Centre for Mathematical Sciences, Lund University, on July 23-24, 2012, and is organized by Johan Öinert (University of Copenhagen) and Sergei Silvestrov (Mälardalen University). You are welcome to call Johan Ö (+46-735-323635) if, for example, you have difficulties entering the building or doors inside the building.

# Schedule

All talks are given in room MH:227 which is located one floor above the ground floor. If you have any problems entering the building or doors inside the building, please call Johan Ö (+46-735-323635) immediately!

## Monday, July 23, 2012

14.00-14.05: Johan Ö & Sergei say Welcome!

14.05-14.50: Johan R talks about Simplicity of Ore extensions

14.50-15.00: Discussion

15.00-15.30: Coffee break in the common room (4th floor)

15.30-16.15: Tron talks about Primitivity and primeness of twisted group C*-algebras

16.15-16.25: Discussion

16.25-16.35: Short break

16.35-17.20: Joakim talks about Poisson Algebraic Geometry and Noncommutative Surfaces

17.20-17.30: Discussion

18.00-**.**: Dinner (in the city centre of Lund)

## Tuesday, July 24, 2012

09.30-10.15: Patrik talks about Degree maps, simplicity and maximal commutativity

10.15-10.25: Discussion

10.25-10.35: Short coffee break (4th floor)

10.35-11.20: Toke talks about Simplicity of Cuntz-Pimsner rings

11.20-11.30: Discussion

11.30-11.35: Short break

11.35-12.20: Johan Ö talks about Partial skew group rings and Leavitt path algebras

12.20-12.30: Discussion

12.30-**.**: Lunch (close to the math building)

# Participants & abstracts of talks

Johan Richter (Lund University, Sweden)
Simplicity of Ore extensions

Abstract:
I will review some conditions for when Ore extensions are simple. Specifically, inspired by results for crossed products, I will discuss whether simplicity has any relation with the following properties: that the coefficient ring is maximal commutative, and that every ideal of the Ore extension intersects the coefficient ring nontrivially. My talk is based on joint work with Johan Öinert and Sergei Silvestrov.

Tron Ånen Omland (Norwegian University of Science and Technology, Norway)
Primitivity and primeness of twisted group C*-algebras

Abstract:
For a multiplier (2-cocycle) \sigma on a discrete group G we give conditions for which the corresponding twisted group C*-algebras are prime or primitive and briefly comment on simplicity and properties on their center. We also discuss simplicity, primeness, primitivity and triviality of center for the twisted (complex) group algebras.

Poisson Algebraic Geometry and Noncommutative Surfaces

Abstract:
In this talk I will discuss particular aspects of Poisson algebras of smooth functions on manifolds and their noncommutative counterparts, in which the commutator of two elements is related to the Poisson bracket of the corresponding smooth functions. In particular, it is shown how classical differential geometry of almost Kähler manifolds can be formulated entirely in terms of Poisson brackets of a certain set of functions on the manifold. Furthermore, as particular and important examples, and also of independent interest, I will describe a generic way to construct noncommutative algebras as analogues of surfaces described as inverse images of polynomials in R^3.

Patrik Nystedt (University West, Sweden)
Degree maps, simplicity and maximal commutativity

Abstract:
For an extension $A/B$ of associative, not necessarily unital rings, we investigate the connection between simplicity of $A$ with a property that we call $A$-simplicity of $B$. By this we mean that there is no non-trivial ideal $I$ of $B$ being $A$-invariant, that is $AI \subseteq IA$. We show that if $A$ is simple, $B$ is a direct summand of $A$ as left $B$-module and every ideal of $B$ has the identity property as a right $B$-module, then $B$ is $A$-simple. To obtain sufficient conditions for simplicity of $A$, we introduce the concept of a degree map $d : A \to \N$ for $A/B$. We show that if the centralizer $C$ of $B$ in $A$ is $A$-simple, every intersection of $C$ with an ideal of $A$ is $A$-invariant, $A C A = A$ and there is a degree map for $A/B$, then $A$ is simple. These results are applied to category graded rings, crossed products, Ore extensions and Cayley doublings of algebras.

Toke Meier Carlsen (Norwegian University of Science and Technology, Norway)
Simplicity of Cuntz-Pimsner rings

Abstract:
Eduard Ortega and I have introduced an algebraic analogue of (relative) Cuntz-Pimsner C*-algebras and shown that for instance Leavitt path algebras and crossed products of a ring by a single automorphism can be constructed as relative Cuntz-Pimsner rings. Recently, Eduard and I have together with Enrique Pardo characterized when a relative Cuntz-Pimsner ring is simple. I will in this talk give an overview of our work and discuss similarities and differences between the algebraic case and the C*-algebraic case.

Johan Öinert (University of Copenhagen, Denmark)
Partial skew group rings and Leavitt path algebras

Abstract:
Partial skew group rings and Leavitt path algebras are two examples of very interesting classes of rings that were introduced during the last decade. In this talk I will state their definitions, give some examples and historical background. Very recently (Feb. 2012) it was shown, by Goncalves & Royer, that Leavitt path algebras can be realized as partial skew group rings. I will state some results (work in progress!) on ideals of partial skew group rings and indicate how this might be used to retain some of Goncalves & Royer's results more efficiently.

Other participants (who are not giving talks) include:
Fredrik Ekström (Lund University, Sweden)
Søren Eilers (University of Copenhagen, Denmark)
Karl Lundengård (Mälardalen University, Sweden)
Sergei Silvestrov (Mälardalen University, Sweden)

Last update: 2012-07-20 @ 10.50 CET