# Download Decompositions of Operator Algebras I and II (Memoirs of the by I. E. Segal PDF

By I. E. Segal

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**Example text**

Now let ) be the range of such a projection 7 is then a two-sided ideal In the every such ideal is invariant under P(JPx) = JPx, or Applying PJP = JP. PJP = P. shows that is conjugate J It is shown in [11] P. 0 I,2-system of in the sense Now J. , JPx a x e 74, Multiplying the last equation on the left by Is SA, so P (PJP)* - P and PPJ = P. to both sides of the last equation shows that J in C P According to a theorem of Ambrose (loc. cit. Th. 7), of Ambrose [2]. J As 3J = 3* for every projection PJ = P linear, it is enough tc show that that SJ = 3*.

On y f Fn. to Let z by setting and the Gn such that Gn g(y), with hi11(y)E be an arbitrary fixed gn(y) = hn(y) Then it is easily seen that y e f(E), y e Fn converges, say to Gn+l' Gn By Sublemma 2 and induction there exists Fn = f(Gn). C, and define point in UUGn, where for sufficiently large g(y) e f 1(y). for gn is n, so It is plain that is a function satisf;ing the conclusion of the lemma. The following lemma shows roughly that the T'/ are irreducible "on the average". 2. 3(r)cr,, where Of ( 70 and 0( 1) are measurable DECOMPOSITIONS OF OPERATOR ALGEBRAS.

G We begin by considering the decomposition of conjugations in We recall that a function suitable situations. to itself is called a conjugation if it satisfies the conditions: 1-f 1) J2 = I, 2) (Jx, Jy) = (y, x) has the properties y on a Hilbert space J in 14 for all x J(x + y) = Jx + Jy, and and complex .. G J(a x) = 67x and that it for all , and that With the notation of Theorem 1, let such that exists for each in H ; x 1( 6 SIT = e for all a conjugation S e C, and T,( on and is a ring (JWJ)* = JW*J.