Summary: EIGENVALUE LISTS OF NONCOMMUTATIVE
Expanded notes for a lecture given 4 May 1999
12 May 1999
Abstract. 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 M2 . . . 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 of B(H) to Mn and considers its eigenvalue list n for large n, then
n should be close to the eigenvalue list of when n is large.
We discuss some natural examples which show that this intuition is wrong, and
we attempt to explain the phenomemon by describing the correct asymptotic formula
when the sequence (Mn) is "stable". Applications are not discussed here, but are
taken up in .
A tower is an increasing sequence of type I subfactors of B(H) M1 M2 . . .
whose union is weak
-dense in B(H). Here H is separable and infinite dimensional,