We construct a functor that maps C*-correspondences to their Cuntz-Pimsner algebras. We then use our functor to investigate the passage of certain C*-correspondence relations to the associated Cuntz-Pimsner algebras. If the time permits, we shall discuss other "possible" applications of this functor.

A categorical approach to C*-algebras arising from C*-correspondences

Thu, Nov. 11 3pm (Zoom)

Probability

Brian Hall (University of Notre Dame)

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The two most basic results in random matrix theory are the circular law and the semicircular law. The circular law says that if N is large and Z is an N x N random matrix with i.i.d. entries of mean zero and variance 1/N, then the eigenvalues of Z will be almost uniformly distributed on the unit disk. The semicircular law says that if N is large and X is a Hermitian random matrix with i.i.d. entries of mean zero and variance 1/N on and above the diagonal, then the (real) eigenvalues of X will have a nearly semicircular distribution on the interval [-2,2].

We now make a simple observation about the relationship between the circular and semicircular laws: twice the real part of the eigenvalues in the circular law has the same limiting distribution as the eigenvalues in the semicircular law. The observation is at one level trivialâ€”if z is uniformly distributed over the unit disk, then 2Re(z) has a semicircular distribution on [-2,2]. But it is, at another level, mysterious: Why should the real parts of the eigenvalues in one random matrix model be related to the eigenvalues in a different random matrix model?

In this talk, I will try to persuade you that the relationship between the circular and semicircular laws is not a coincidence but part of a general phenomenon relating the bulk eigenvalue distributions of different random matrix models. For example, if X0 is an arbitrary Hermitian random matrix independent of Z and X, then there is a formula relating the real parts of the eigenvalues of X0+Z to the eigenvalues of X0+X. The talk will be self-contained and have lots of pictures.