Nonexistence of a normal conditional expectation in a continuous crossed product
Yoshikazu Katayama · Kodai Mathematical Journal · 1981
KATAYAMA 1. Introduction.The conditional expectations of operator algebras played an important role from the outset in the theory of operator algebras.J. Diximeir [3] and H. Umegaki [14] have introduced conditional expectations in a finite von Neumann algebra onto its von Neumann subalgebras and there are abundant systematic studies concerning the conditional expectations (See for example [3],.[10], -, [17]).Besides, we have the notion of a crossed product.It is constructed from a triple (M, G, a) where M is a von Neumann algebra, G is a locally compact group and a is an action of G on M, i.e. α is a homomorphism of G into the automorphism group of M satisfying certain continuity conditions.We call it a W*-dynamical system.The method of construction of the crossed product Gx a M from a PF*-dynamical system (M, G, a) will be made explicit in § 2.. Further we will call it a discrete crossed product when G is a discrete group, and a continuous crossed product when G is not discrete.Now, in the case of a discrete crossed product, there exists a faithful normal conditional expectation of Gx a M onto M.But it was not known, in the case of a continuous crossed product, whether there exists a normal conditional expectation of Gx a M onto M.In this note we establish the following theorem; There is no normal conditional expectation of Gx a M onto M if G is a locally compact connected group and if there is an element h in G such that a h is an outer automorphism of M.In spite of this result, a normal semi-finite operator valued weight from a crossed product Gx a M into M can always be found.This was shown by IL Haagerup [4] prior to our result. Notations and Preliminaries. Let M be a von Neumann algebra on aHubert space H and G be a locally compact group.The triple (M, G, a) is said a T^*-dynamical system if the mapping a of G into the group Aut(M) of all automorphisms of M is a homomorphism and the function g-^ωa g (x) is continuous on G for any XGM and ωεM^ (M* is the predual of M).The crossed product Gx a M of M with G is the von Neumann algebra on