A COMMUTATIVE AND COMPACT DERIVATIONS FOR W* ALGEBRAS

In this paper, we study the compact derivations on W* algebras. Let M be W*-algebra, let ( ) LS M be algebra of all measurable operators with M , it is show that the results in the maximum set of orthogonal predictions. We have found that W* algebra A contains the Center of a W* algebra ß and is either a commutative operation or properly infinite. We have considered derivations from W* algebra two-sided ideals.


INTRODUCTION
Let M be a W*-algebra and let ( ) The norm closing two sided ideal ( ) fB generated by the finite projections of a W* algebra B behaves somewhat similar to the idealized compact operators of ( ) BH (see [11], [8], [9]).
Therefore, it is natural to ask about any sub-algebras d of B that is any derivation from  into ( ) fB implemented from an element of ( ) yB.
Int. J. Anal. Appl. 18 (4) (2020) 645 We perform two main difficulties: the presence of the center of B and the fact that the main characteristic in [8] Since 0 FG = , by the maximally of the family we have ( ) TF   .

A COMMUTATIVE OPERATION ON W* SUB-ALGEBRAS
When A a commutative operation is is crucial because it provides the following explicit way to find an operator TB  implementing the derivation.
For the rest of this section let Thus M is bounded and for all f (see [8] for the existence and We have seen that given an invariant mean . We collect several properties of   Q .

Corollary (8). Let
B be a semi-finite W* algebra with a trace  , let  be a properly infinite W* sub-algebras of B and let 11  +   . Then for every derivation In the notations introduced there, it is easy to see that completes the proof.

Corollary (10). (i) If
Ep  then the following hold: Proof. We have to show that for every (v) Follows at once from (ii) and (iv).
The condition that   In addition, and is properly infinite projection. Hence, in the case when is finite projection, it follows that . Let us consider the case when is a properly infinite projection with and such that . In this case, with and deduce .
All other statements follow from the form of element . Since, and for every . Observe also that .
Finally, let all projections be finite. Since , we have for every . There projections standing on the right-hand sides are finite. Hence, is finite projection as a sum of the left-hand sides [22].
We shall use a following well-known implication .
We supply here a straightforward argument. Let be such that . Then and therefore . This means .
As in [6] we can use Theorem (6) to extend the result to the properly infinite case.
Theorem (13). Let  be a properly infinite W* sub-algebra of B containing the center of B .
For every derivation Before we start the proof let us recall that if  is properly infinite there is an infinite countable decomposition of the identity into mutually orthogonal projections of  , all ( ) ( ) ( ) ( ) ( ) where the latter is the non commutative 1 L  + -space of B relative to  (see [14]).
Recall the following facts about      No data were used to support this study.