RELATIVE ENTROPY FOR MAXIMAL ABELIAN SUBALGEBRAS OF MATRICES AND THE ENTROPY OF UNISTOCHASTIC MATRICES

Abstract

Let A and B be two maximal abelian *-subalgebras of the n × n complex matrices Mn(ℂ). To study the movement of the automorphisms of Mn(ℂ), we modify the Connes–Størmer relative entropy H(A|B) and the Connes relative entropy Hϕ(A|B) with respect to a state ϕ, and introduce the two kinds of the constant h(A|B) and hϕ(A|B). For the unistochastic matrix b(u) defined by a unitary u with B = uAu*, we show that h(A|B) is the entropy H(b(u)) of b(u). This is obtained by our computation of hϕ(A|B). The h(A|B) attains to the maximal value log n if and only if the pair {A, B} is orthogonal in the sense of Popa.