Entropy for a Pair of Subalgebras via Automorphisms

This is an expository survey article based on a talk for the Special Session (Operator Algebra) in AMC 2009. For two subalgebras A and B of a finite von Neumann algebra M, we discuss a modified version h(A│B) of the Connes-St rmer relative entropy H(A│B): As one of the most elementary examples, we pick two maximal abelian *-subalgebras A│B of the n × n complex matrices Mn(C); and show relations between h(A│B) and the entropy for unistochastic matrices. Also we show related results in the case of a pair of subfactors of a type II1 factor.