Eigenvalue Lists of Noncommutative Probability Distributions
by
William Arveson
University of California, Berkeley

In probability theory all nonatomic probability measures look the same. That is because any two nonatomic separable measure algebras are isomorphic. Quantum probability theory is different: two normal states of B(H) are conjugate only when the eigenvalue lists of their density operators are the same. Suppose now that one is given an increasing sequence M₁ subset or equal M₂ subset or equal ... of type I subfactors of B(H) whose union is weak^*-dense in B(H). Common sense suggests that if one restricts a normal state \rho of B(H) to M_n and considers its eigenvalue list \Lambda_n for large n, then \Lambda_n should be close to the eigenvalue list of \rho when n is large.

We discuss some natural examples which show that this intuition is wrong, and we attempt to explain the phenomenon by describing the correct asymptotic formula when the sequence (M_n) is ``stable". Time permitting, we describe the problem in noncommutative dynamics that led to these issues, and give applications.

Date received: May 17, 1999