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)
Brian Hall (University of Notre Dame)
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 trivial—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.