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M-COMPLETE APPROXIMATE IDENTITIES IN OPERATOR SPACES
 

Summary: M-COMPLETE APPROXIMATE
IDENTITIES IN OPERATOR SPACES
A. Arias and H.P. Rosenthal
Abstract. This work introduces the concept of an M-complete approx-
imate identity (M-cai) for a given operator subspace X of an operator
space Y . M-cai's generalize central approximate identities in ideals in
C -algebras, for it is proved that if X admits an M-cai in Y , then X
is a complete M-ideal in Y. It is proved, using \special" M-cai's, that
if J is a nuclear ideal in a C -algebra A, then J is completely com-
plemented in Y for any (isomorphically) locally re exive operator space
Y with J Y A and Y=J separable. (This generalizes the previ-
ously known special case where Y = A, due to E ros-Haagerup.) In
turn, this yields a new proof of the Oikhberg-Rosenthal Theorem that K
is completely complemented in any separable locally re exive operator su-
perspace, K the C -algebra of compact operators on `2. M-cai's are also
used in obtaining some special a rmative answers to the open problem of
whether K is Banach-complemented in A for any separable C -algebra A
with K A B(`2). It is shown that if conversely X is a complete M-
ideal in Y, then X admits an M-cai in Y in the following situations: (i) Y
has the (Banach) bounded approximation property (ii) Y is 1-locally re-

  

Source: Arias, Alvaro - Department of Mathematics, University of Denver

 

Collections: Mathematics