Matthew Lorentz

Abstract:
Many times in analysis we focus on the “small scale” structure of a metric space, e.g. continuity, derivations, etc. However, to examine the “large scale” structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X, dX) is coarsely equivalent to (Y, dY ) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely bornological definitions exist). In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules.

Abstract:
Many times in analysis we focus on the “small scale” structure of a metric space, e.g. continuity, derivations, etc. However, to examine the “large scale” structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X, dX) is coarsely equivalent to (Y, dY ) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely bornological definitions exist). In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules.

Abstract:
In order to help us better understand the structure of a space we look for invariants, not only of the space but also invariants of its algebra of functions. One such invariant is the Hochschild (co)homology. Using the Hochschild-Kostant-Rosenberg theorem (for sufficiently well behaved commutative algebras) one may identify the Hochschild homology with differential forms and the cohomology with multivector fields. Thus, for a noncommutative algebra we may consider its Hochschild (co)homology as noncommutative analogs of differential forms and multivector fields respectively. Many times in analysis we focus on the ``small scale" structure of a metric space, e.g., continuity, derivations, etc. However, to examine the ``large scale" structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X,d_X) is coarsely equivalent to (Y, d_Y) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely topological definitions exist). In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules.

Abstract:
Many times in analysis we focus on the ``small scale" structure of a metric space, e.g., continuity, derivations, etc. However, to examine the ``large scale" structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space (X,d_X) is coarsely equivalent to (Y, d_Y) then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable C*-algebra contained in the bounded operators on square summable sequences indexed by a metric space X (note that purely topological definitions exist). In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules.

Abstract:
In Rufus Willett's and my paper "Bounded Derivations on Uniform Roe Algebras" we showed that all bounded derivations on a uniform Roe algebra associated to a bounded geometry metric space are inner. This naturally leads to the question of whether or not the higher dimensional Hochschild cohomology groups of the uniform Roe algebra vanish also. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Lastly, we show that if the norm continuous Hochschild cohomology of a uniform Roe algebra vanishes in all dimensions then the ultraweak-weak* continuous Hochschild cohomology of that uniform Roe algebra vanishes also.

Abstract:
In Rufus Willett's and my paper "Bounded Derivations on Uniform Roe Algebras" we showed that all bounded derivations on a uniform Roe algebra associated to a bounded geometry metric space are inner. This naturally leads to the question of whether or not the higher dimensional Hochschild cohomology groups of the uniform Roe algebra vanish also. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Lastly, we show that if the norm continuous Hochschild cohomology of a uniform Roe algebra vanishes in all dimensions then the ultraweak-weak* continuous Hochschild cohomology of that uniform Roe algebra vanishes also.

Abstract:
In Rufus Willett's and my paper "Bounded Derivations on Uniform Roe Algebras" we showed that all bounded derivations on a uniform Roe algebra associated to a bounded geometry metric space are inner. This naturally leads to the question of whether or not the higher dimensional Hochschild cohomology groups of the uniform Roe algebra vanish also. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Lastly, we show that if the norm continuous Hochschild cohomology of a uniform Roe algebra vanishes in all dimensions then the ultraweak-weak* continuous Hochschild cohomology of that uniform Roe algebra vanishes also.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.