The Tomita Decomposition of Rings of Operators

Joseph L. Taylor
1964 Transactions of the American Mathematical Society  
i) I. Introduction. It is known that if R is a symmetric ring of bounded operators on a separable Hubert space H and f 0 is a vector in H which is cyclic with respect to R, then the positive functional F(A)= (AÇ0, £0), for A e R, may be written as a direct integral over a compact Hausdorff space M, i.e., F(A) = ¡Mfm(A)dp(m) where p is a positive regular Borel measure and the functionals fm are indecomposable except, at most, for m eM0 <= M and p(M0) -0. This decomposition of F induces a
more » ... F induces a representation of R as a direct integral of rings Rm of operators on a Hubert space Z/m and for almost all m (mod p), Rm is an irreducible ring on Hm. The problem of extending this type of decomposition to rings of operators on an arbitrary Hubert space was attacked in 1954byTomita (cf. [6]) using extremely penetrating techniques. However, certain parts of Tomita's development of his decomposition theory require a special measure theoretic result which is not valid in general. Consequently, the question of whether or not this measure theoretic difficulty could be circumvented arose ; i.e., did the Tomita decomposition hold for arbitrary rings and, if not, for what type of rings did it hold? In this paper we shall first show (Theorem 2.3) that in case R is a weakly closed symmetric ring which contains its commutant, then the Tomita decomposition
doi:10.2307/1994089 fatcat:snq25yvntrgdjniyuoiuxmv42e