A short proof of a decomposition theorem of a von Neumann algebra

Abstract

Abstract. Let M be a von Neumann algebra and 5 and T be commuting "-automorphisms on M satisfying the equation: 5 + S"1 = T + T~l. It is proved that M can be decomposed by a central projection p in M such that S = T on Mp and S= T-1 on M(\ -p).

A NOTE ON Α-Derivations ON SEMIPRIME RINGS

“…[1] where further references can be found. We refer to Herstein [6] for the general theory of rings.…”

On certain pairs of automorphisms of C*-algebras

“…The following theorem gives conditions which are both necessary and sufficient for a pair of ^-automorphisms to satisfy (1). Although the conditions are not entirely intrinsic to the C* -algebra A (they involve the weak closure of A in a certain representation), they imply certain other conditions which are intrinsic, necessary, and close to being sufficient (see the subsequent corollaries).…”

“…The proof depends on an inspection of the weak closure of A in its atomic representation, which is a direct sum of type I factors. Preliminary results for type I factors, giving necessary and sufficient conditions for (1) and similar equations, are given in Section 2. In Section 4, it is shown how the same methods lead to proofs of the corresponding results for one-parameter groups of automorphisms.…”

“…The identity p of the ultraweak closure of 7 2 w.^tf then satisfies j3t\-^P = The first part of the proof of Theorem 4.1 needs only very slight modification to show the commutativity of weakly continuous actions satisfying (2).…”

“…The equation (1) a + a ' 1 =/? + / T It is easily seen that (1) implies that a" + a~n -/3 n + P~n for all integers n.…”