The Weyl group and the normalizer of a conditional expectation (Q1300156)

Let \(N\) and \(M\), \(N\subset M\), be von Neumann algebras; \(E: M\to N\) a conditional expectation; \(N_E\) the normalizer of \(E\). The authors define a discrete group \(W(E)\), which they call the Weyl group of \(E\), associated to a faithful normal \(E\), which shows the relation between the group \(N_E\) and the unitary group \(U_N\). It is shown that the group \(W(E)\) is trivial if \(N\) and \(M\) are \(\text{II}_1\) factors and \(\text{Ind}(E)< 4\); \(W(E)\) is finite if \(\dim Z(N)< \infty\) and bounded by the index in the factor case. When \(M\) is the crossed product of \(N\) by a property outer discrete group \(G\) of automorphisms, \(W(E)\) contains the original group, i.e. \(G\subset W(E)\). In general \(W(E)\) is bigger than \(G\), also when \(G\) is finite. Nevertheless, it is proved that if \(G\) is finite and \(N\) is a factor, then \(G\cong W(E)\). Moreover, the authors get the conditions for the finiteness of \(W(E)\) and bounds for its order.