Index for pairs of finite von Neumann algebras
The Jones' index of a pair (N,M) of finite von Neumann algebras with finite dimensional centres has been given two definitions: one ring- theoretic, and one using Markov traces. We extend the second definition to the case of finite, countably decomposable von Neumann algebras and we show that the two definitions agree when the algebras are direct sums of finite factors. We also study Markov traces on such pairs. More precisely: When N is of finite index in M, i.e. if M acts on some Hilbert space H in such a way that the commutant \(N'_ H\) of N is finite, and that the coupling operators \(c_ M(H)^{\pm 1}\) and \(c_ N(H)^{\pm 1}\) are bounded, then we define two bounded, linear, normal maps \(C^ M_ N\) in \(L_*(Z(N))\) and \(D^ M_ N\) in \(L_*(Z(M))\) in terms of the coupling operators and the centre-valued traces, which do not depend on the chosen representation and which have the same spectral radius. The index of N in M, denoted by [M:N], is then the common spectral radius of \(C^ M_ N\) and of \(D^ M_ N\).
finite von Neumann algebras with finite dimensional centres
finite, countably decomposable von Neumann algebras
scientific article; zbMATH DE number 4084358