 
Summary: MCOMPLETE APPROXIMATE
IDENTITIES IN OPERATOR SPACES
A. Arias and H.P. Rosenthal
Abstract. This work introduces the concept of an Mcomplete approx
imate identity (Mcai) for a given operator subspace X of an operator
space Y . Mcai's generalize central approximate identities in ideals in
C algebras, for it is proved that if X admits an Mcai in Y , then X
is a complete Mideal in Y. It is proved, using \special" Mcai'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 rosHaagerup.) In
turn, this yields a new proof of the OikhbergRosenthal Theorem that K
is completely complemented in any separable locally re exive operator su
perspace, K the C algebra of compact operators on `2. Mcai's are also
used in obtaining some special a rmative answers to the open problem of
whether K is Banachcomplemented 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 Mcai in Y in the following situations: (i) Y
has the (Banach) bounded approximation property (ii) Y is 1locally re
