September 10 
Chris Ramsey, UVaOperator algebras for analytic varietiesTo an analytic variety V one can consider the restriction of the multiplier algebra of DruryArveson space to V. The goal of these two talks is to show that two such operator algebras are completely isometrically isomorphic if and only if there is an automorphism of the ball taking one variety onto the other. 
September 17 
Chris Ramsey, UVaOperator algebras for analytic varietiesTo an analytic variety V one can consider the restriction of the multiplier algebra of DruryArveson space to V. The goal of these two talks is to show that two such operator algebras are completely isometrically isomorphic if and only if there is an automorphism of the ball taking one variety onto the other. 
September 24 
Scott Atkinson, UVaBasic notions of free probability (Part 1 of 2)In Part 1, we will first set up the basic framework of noncommutative probability and discuss some basic notions of this topic with examples. The notion of free independence will be introduced and contrasted with classical (tensor) independence. To illustrate this contrast, we will exhibit combinatorial proofs of the Central Limit Theorem when either of these two notions of independence is considered. Looking ahead to Part 2, there are a number of possible topics we may discuss, including but not limited to: computations using the free central limit theorem, the surprising result of Wigner's theorem for large random matrices, the relation between free independence and noncrossing partitions via free cumulants, free convolution, and the Rtransform. 
October 1 
Scott Atkinson, UVaBasic notions of free probability (Part 2 of 2)We will begin by completing the proof of the Central Limit Theorems in this algebraic context. As we will see in this proof, noncrossing partitions of n arise naturally from free independence. We will go on to examine this relationship between noncrossing partitions and free independence by looking at the concept of free cumulants (related to moments via the momentcumulant formula). These free cumulants will let us study free convolution and the related Rtransform (this is the free analog of the logarithm of the Fourier transform). We will use these ideas to discuss the free convolution of two projections and the free analog of the Poisson distribution. 
October 8 
Department reception for Barbara and Tom in the lounge (3:30 pm)October 12 is VOTCAM, in honor of Barbara and Tom, at UVa. 
October 15 
NO MEETING  Reading Day 
October 22 
Kostya Medynets, US Naval AcademyCharacters and II_{1} Factor Representations of Thompson's groupA character of an infinite group G is defined as a positivedefinite class function. In view of the GelfandNaimarkSegal construction, characters are in onetoone correspondence with finite (in the sense of Murrayvon Neumann) von Neumann algebra representations of the group. Characters of a group form a Choquet simplex with extreme characters being in onetoone correspondence with type II_1 factor representations of the group in question.Studying the representation theory of the infinite symmetric group S(∞), Vershik noticed that many of its characters come from ergodic actions of S(∞) on measure spaces (X,μ) by the formula f(g) = μ(FixedPoints(g)). The question we are trying to address is for which infinite groups all of the characters can be given a similar dynamical interpretation. We will reexamine some older results on classification of characters of such groups as Special Linear Group of Infinite Matrices over a finite field and Groups of Rational Permutation of the unit interval. We will then classify characters of locally finite groups determined by Bratteli diagrams and the HigmanThompson groups. In the case of (the commutator of) a HigmanThompson group, based on the fact that its natural action on the unit interval has no invariant measures, we will show that this group has no nontrivial characters. As a result, the only nontrivial II_{1} factor representation of it is the group von Neumann algebra. The absence of nontrivial characters has some implications in the theory of random subgroups. 
October 29 
David Sherman, UVaEquivalent operator categoriesLeaving rigorous definitions to the talk, operator categories are natural classes that include C*algebras, operator systems, hereditary manifolds, operator algebras, Jordan operator algebras, etc. I will show how to associate the following features to any such category: an operator topology, a representation theory, and a convexity/dilation theory. It turns out that if one of these features agrees for a pair of categories, then all three do, in which case the categories are called equivalent. I will discuss some equivalences, along the way obtaining new observations about Arveson's hyperrigidity and maybe even triangles. 
November 5 
Jocelyn Xue, University of RichmondEvery matrix is a product of Toeplitz matricesRecently Ye and Lim showed that every matrix can be written as a product of Toeplitz matrices. In this talk, I will outline a proof of this result. I will also discuss the maximal algebras within the Toeplitz matrices and show how some alterations of the Ye and Lim proof will show that the n by n matrices can be written as a product of maximal algebras of Toeplitz matrices. 
November 12 
Brian Lins, HampdenSydney CollegeInverse continuity of the numerical range mapThe numerical range of a matrix has many applications in linear algebra, physics, and engineering. The numerical range is the image of the complex unit nsphere under the quadratic map [x maps to x*Ax]. Recently there has been interest in the preimages of this numerical range map. I will discuss the continuity properties of the multivalued inverse of the numerical range map. Along the way, we will revisit some of the classical results concerning the numerical range. 
November 19 
Sean Cox, VCUForcing Axioms and Baire Category TheoremThe Baire Category Theorem (BCT) states that in certain topological spaces, the intersection of countably many open dense sets is dense. "Forcing Axioms" assert that for a wide class of Stone Spaces, the "countably many" requirement in the BCT can be relaxed. Forcing Axioms can neither be proved nor disproved in classical mathematics, but have many interesting consequences; e.g. Farah proved that Forcing Axioms imply there are no outer automorphisms of the Calkin algebra. I will discuss Forcing Axioms and their enemy, Jensen's Diamond Principle. 
November 26 
Stephen Hardy, UVaNaimark's problem: a primerWe know that every irreducible representation of the compact operators on a Hilbert space is unitarily equivalent to the identity representation. In 1951, Naimark asked if the converse was true, and this came to be known as Naimark's problem. We will discuss the history of Naimark's problem, including results of Rosenberg and Glimm. We will discuss a criterion for unique extensions of pure states due to Anderson and then introduce an important crossed product construction utilizing the homogeneity of the state space of separable C*algebras (Futamura, Kataoka, and Kishimoto 2001, Kishimoto, Ozawa, and Sakai 2003). In a subsequent talk we will present the 2004 construction of a counterexample to Naimark's problem due to Akemann and Weaver. 
December 12

Stephen Hardy, UVaNaimark's problem: the counterexampleIn a previous talk we discussed the history of Naimark's problem. In this talk we will work through the construction of a counterexample to Naimark's problem: a C*algebra with only one irreducible representation up to unitary equivalence which is not *isomorphic to the compact operators on a Hilbert space. This construction is due to Akemann and Weaver in 2004 and utilizes Jensen's diamond principle, which is known to be independent of ZFC. 