Abstract:

In this talk I will show that for uniform Roe algebra associated to a bounded geometry metric space, that all bounded derivations from that uniform Roe algebra to itself are inner. We obtain this result using a “reduction of cocycles" method from Sinclair and Smith. Then the key technical ingredient comes from recent work of Braga and Farah in their paper “On the Rigidity of Uniform Roe Algebras".
That all bounded derivations are inner is equivalent to the first Hochschild cohomology group vanishing. It is then natural to ask if all the higher Hochschild cohomology groups vanish. I will define the Hochschild cohomology and explain the current obstacles to adapting Sinclair and Smith's methods to our situation.

## Matthew Lorentz

Graduate Student

University of Hawai‘i at Mānoa

__Curriculum Vitae__

Welcome! I am currently a graduate student at the University of Hawai‘i at Mānoa studying under Rufus Willett. My interests are primarily C*-algebras and K-theory.

### Talks

Bounded Derivations and the Hochschild Cohomology of Uniform Roe Algebras.

Washington University St. Louis Analysis Seminar, 04-5-2021

Bounded Derivations and the Hochschild Cohomology of Uniform Roe Algebras.

ASU C*-Seminar, 02-3-2021

Abstract:

In this talk I will show that for uniform Roe algebra associated to a bounded geometry metric space, that all bounded derivations from that uniform Roe algebra to itself are inner. We obtain this result using a “reduction of cocycles" method from Sinclair and Smith. Then the key technical ingredient comes from recent work of Braga and Farah in their paper “On the Rigidity of Uniform Roe Algebras".
That all bounded derivations are inner is equivalent to the first Hochschild cohomology group vanishing. It is then natural to ask if all the higher Hochschild cohomology groups vanish. I will define the Hochschild cohomology and explain the current obstacles to adapting Sinclair and Smith's methods to our situation.

The Hochschild Cohomology of Uniform Roe Algebras.

Online Noncommutative Geometry Seminar WUSTL, 10-28-2020

Abstract:

Recently Rufus Willett and I showed that all bounded derivations on Uniform Roe Algebras associated to a bounded geometry metric space \(X\) are inner in our paper "Bounded Derivations on Uniform Roe Algebras".
This is equivalent to the first Hochschild cohomology group \(H^1(C_u^*(X),C_u^*(X))\) vanishing. It is then natural to ask if all the higher groups \(H^n(C_u^*(X),C_u^*(X))\) vanish.
To investigate the continuous cohomology of a Uniform Roe Algebra we employ the technique of "reduction of cocycles" where we modify a given cocycle by a coboundary to obtain certain properties.
I will discuss this procedure and give examples of calculating the higher cohomology groups.

The Hochschild Cohomology of Uniform Roe Algebras.

University of Wollongong Operator Algebras and Noncommutative Geometry Seminar, 7-23-2020

Abstract:

Recently Rufus Willett and I showed that all bounded derivations on Uniform Roe Algebras associated to a bounded geometry metric space \(X\) are inner in our paper "Bounded Derivations on Uniform Roe Algebras".
This is equivalent to the first Hochschild cohomology group \(H^1(C_u^*(X),C_u^*(X))\) vanishing. It is then natural to ask if all the higher groups \(H^n(C_u^*(X),C_u^*(X))\) vanish.
To investigate the continuous cohomology of a Uniform Roe Algebra we employ the technique of "reduction of cocycles" where we modify a given cocycle by a coboundary to obtain certain properties.
I will discuss this procedure and give examples of calculating the higher cohomology groups.

Bounded Derivations on (Not Necessarily Nuclear) Uniform Roe Algebras.

Conference: The 48th Canadian Operator Symposium, The Fields Institute, 5-25-2020

Abstract:

In this talk we prove that for a uniform Roe algebra associated to a bounded geometry metric space all bounded derivations on that uniform Roe algebra are inner derivations.

Bounded Derivations on (Not Necessarily Nuclear) Uniform Roe Algebras.

Conference: C*-Algebras and K-Theory, University of Hawai'i, 12-5-2019

Abstract:

In this talk we prove that for a uniform Roe algebra associated to a bounded geometry metric space all bounded derivations on that uniform Roe algebra are inner derivations.

The Gelfand-Naimark-Segal Construction.

University of Hawai'i AMS chapter Graduate Student Seminar, 10-17-2019

Abstract:

In this talk I will summarize how to build representations from a C*-algebra to B(H) (the bounded operators on a Hilbert space) using the GNS construction. Just like we use homomorphisms to study groups and rings, we can use *-homomorphisms to study *-algebras . Moreover, if the target space is linear we can use this to further understand the behavior of our *-algebra. The GNS construction allows us to build a wealth of *-homomorphisms to the linear space B(H) via representations. Hopefully this talk will be accessible to anyone with a basic understanding of linear and abstract algebra.

Bounded Derivations on Nuclear Uniform Roe Algebras.

University of Hawai’i Noncommutative Geometry Seminar 8-30-19, 9-6-19, and 9-20-19

Abstract:

In this series of talks we give conditions on a space X to give a positive answer to the question of whether or not all the derivations of the uniform Roe algebra on a space X are inner; that is, if the derivation is given by the commutator bracket [ ,b]. Specifically, if a space X has a metric d under which (X,d) is a metric space with bounded geometry having property A, then all derivations are inner. In the first talk I will reduce the problem to a simpler question. Then show that this new question can be partially answered using the paper of Spakula and Tikuisis that was discussed in our seminar last spring. If we have time I will give an overview of the material contained in their paper.

Abstract:

In this series of talks we give conditions on a space X to give a positive answer to the question of whether or not all the derivations of the uniform Roe algebra on a space X are inner; that is, if the derivation is given by the commutator bracket [ ,b]. Specifically, if a space X has a metric d under which (X,d) is a metric space with bounded geometry having property A, then all derivations are inner. In the second talk I will state the main theorem from the paper of Spakula and Tikuisis that we will need. Then the rest of the talk will be focused on results due to Braga and Farah from their paper “On the Rigidity of Uniform Roe Algebras”. These results will allow us to consider certain families of operators simultaneously. That is, for these families, given epsilon there exists an R such that every member of this family is within epsilon of an operator of propagation at most R.

Abstract:

In this series of talks we give conditions on a space X to give a positive answer to the question of whether or not all the derivations of the uniform Roe algebra on a space X are inner; that is, if the derivation is given by the commutator bracket [ ,b]. Specifically, if a space X has a metric d under which (X,d) is a metric space with bounded geometry having property A, then all derivations are inner. For The last talk we will prove the main theorem; that is, we will show that if X is a metric space with bounded geometry having property A then all bounded derivations on the Uniform Roe algebra of X are inner.

A Generalized Dimension Function for \(K_0\) of C*-algebras.

University of Hawai’i Noncommutative Geometry Seminar 11-21-18

Abstract:

One way to classify projections in the matrices over the complex numbers is to consider their rank. Since the complex numbers are the prototypical C*-algebra we would like to generalize this to C*-algebras to help us better understand their structure and perhaps assist with their classification. Note that matrices of different sizes can have the same rank. Thus, to consider matrices of all sizes we use K-theory, denoted K_0(A), the K-theory of the C*-algebra A. Recall that we can compute the rank of a square idempotent matrix over the complex numbers by taking its trace. We can generalize this to many C*-algebras; however, sometimes we cannot. To this end, we will define and use an unbounded trace. Then using our unbounded trace we will create a group homomorphism from K_0(A) to the complex numbers viewed as an additive group.

G-C*-Algebras.

University of Hawai’i Noncommutative Geometry Seminar 9-19-18

Abstract:

We will discuss groups acting on compact Hausdorff spaces by homeomorphism and C*-algebras by automorphism. This will lead to showing that P(X) is weak*-closed and G-invariant. Lastly we will give an introduction to a G-boundary.

### Papers

Bounded Derivations on Uniform Roe Algebras

Joint work with Rufus Willett

Rocky Mountain J. Math. 50 (2020), no. 5, 1747--1758. doi:10.1216/rmj.2020.50.1747.

Bounded Derivations on Nuclear Uniform Roe Algebras

(Short version) with Rufus Willett

Bounded Derivations on Nuclear Uniform Roe Algebras

(Long version)

A Generalized Dimension Function for \(K_0\) of C*-Algebras

(For partial fulfillment of my specialty exam)

### Conferences / Workshops Attended

Groundwork for Operator Algebras Lecture Series

Role: Teaching Assistant, June 6 - July 19 2020

C*-Algebras and K-Theory, University of Hawai'i

December 2-6, 2019

YMC*A / YWC*A - Københavns Universitet

August 5-12, 2019

Operator Algebras, Groups and Quantum Information - Instituto de Ciencias Matemáticas

June 17-21, 2019

AMS Spring Central and Western Joint Sectional Meeting - University of Hawai'i at Mānoa

March 22-24, 2019

YMC*A / YWC*A - KU Leuven

August 11-17, 2018

### Contact Me

lorentzm [at] math.hawaii.edu