| Talk_id | Date | Speaker | Title |
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29110
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Tuesday 9/14
11:10 AM
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Matthew Lorentz, Michigan State University
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Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 1
- Matthew Lorentz, Michigan State University
- Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 1
- 09/14/2021
- 11:10 AM - 12:00 PM
- C304 Wells Hall
- Brent Nelson (banelson@msu.edu)
In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map.
In our first talk we will begin defining and discussing derivations as they are an important building block of Hochschild cohomology. Motivated by the needs of mathematical physics and the study of one-parameter automorphism groups, it is interesting to study whether all derivations are inner (i.e. given by the commutator bracket) for a particular C*-algebra. In the 1970s, a complete solution to this problem was obtained in the separable case via the work of several authors. For non-separable C*-algebras the picture is murkier. Our main goal in this talk is to give a new class of examples that only have inner derivations: uniform Roe algebras, which are separable only in the trivial finite dimensional case. Uniform Roe algebras were originally introduced for index-theoretic purposes but are now studied for their own sake as a bridge between C*-algebra theory and coarse geometry, as well as having interesting applications to single operator theory. Lastly, we will briefly explain how the uniform Roe algebra only having inner derivations is equivalent to the first Hochschild cohomology vanishing.
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29131
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Tuesday 9/21
11:10 AM
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Matthew Lorentz, Michigan State University
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Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 2
- Matthew Lorentz, Michigan State University
- Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 2
- 09/21/2021
- 11:10 AM - 12:00 PM
- C304 Wells Hall
- Brent Nelson (banelson@msu.edu)
In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map.
In our second talk we will continue discussing derivations as they are an important building block of Hochschild cohomology. Motivated by the needs of mathematical physics and the study of one-parameter automorphism groups, it is interesting to study whether all derivations are inner (i.e. given by the commutator bracket) for a particular C*-algebra. In the 1970s, a complete solution to this problem was obtained in the separable case via the work of several authors. For non-separable C*-algebras the picture is murkier. Our main goal in this talk is to give a new class of examples that only have inner derivations: uniform Roe algebras, which are separable only in the trivial finite dimensional case. Uniform Roe algebras were originally introduced for index-theoretic purposes but are now studied for their own sake as a bridge between C*-algebra theory and coarse geometry, as well as having interesting applications to single operator theory. We will then briefly explain how the uniform Roe algebra only having inner derivations is equivalent to the first Hochschild cohomology vanishing. Lastly, we will discuss the Hochschild cohomology in higher dimensions.
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29139
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Tuesday 9/28
11:10 AM
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Brent Nelson, Michigan State University
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Tomita–Takesaki theory for von Neumann algebras
- Brent Nelson, Michigan State University
- Tomita–Takesaki theory for von Neumann algebras
- 09/28/2021
- 11:10 AM - 12:00 PM
- C304 Wells Hall
- Brent Nelson (banelson@msu.edu)
Tomita–Takesaki theory is a powerful (and often necessary) tool for studying von Neumann algebras that lack tracial states. It shows that any state on a von Neumann algebra $M$ automatically induces an action $\mathbb{R}\curvearrowright M$ (i.e. a non-commutative dynamical system) that then allows one to construct a crossed product von Neumann algebra $M\rtimes \mathbb{R}$. As the notation suggests, this crossed product is an analogue of semidirect products for groups, and it both contains $M$ and encodes the action via unitary elements. It turns out this crossed product always admits a trace (albeit an infinite one) and a dual action $\mathbb{R}\curvearrowright (M\rtimes \mathbb{R})$, and this structure was used by Alain Connes in 1973 to give a classification of the so-called type III von Neumann algebras. In this expository talk, I will provide an introduction to these ideas that does not require any previous experience with von Neumann algebras aside from some functional analysis (e.g. Hilbert spaces, bounded operators, and dual spaces).
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29153
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Tuesday 10/12
11:10 AM
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Matthew Lorentz, Michigan State University
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Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 3
- Matthew Lorentz, Michigan State University
- Derivations and the Hochschild Cohomology of Uniform Roe Algebras, Part 3
- 10/12/2021
- 11:10 AM - 12:00 PM
- C304 Wells Hall
- Brent Nelson (banelson@msu.edu)
In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map.
Hochschild cohomology was introduced by Gerhard Hochschild in his 1945 paper “On the Cohomology Groups of an Associative Algebra”. The Hochschild cohomology of associative algebras has become a useful object of study in many fields of mathematics such as representation theory, mathematical physics, and noncommutative geometry, to name a few.
Last time we showed that all bounded derivations on uniform Roe algebras are inner. The first Hochschild cohomology measures how close derivations are to being inner. Hence, our result from last time can be restated as the first Hochschild cohomology of the uniform Roe algebra vanishing. It is then natural to ask if the higher dimensional cohomologies also vanish.
We will begin with the definition and several properties of multilinear maps which are essential to building the Hochschild complex. We then define the Hochschild complex and Hochschild cohomology as they apply to multilinear maps from a C*-algebra A to a Banach A-bimodule V. We then review many properties of these cohomologies. Lastly, we will show that if all n-linear maps have a weakly continuous representation in the Hochschild cohomology then the n’th dimensional Hochschild cohomology vanishes.
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