M-Complete approximate identities in operator spaces
Abstract
This work introduces the concept of an M-complete approximate 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 complemented in Y for any (isomorphically) locally reflexive operator space Y with
J⊂Y⊂A
and
Y/J
separable. (This generalizes the previously known special case where
Y=A
, due to Effros-Haagerup.) In turn, this yields a new proof of the Oikhberg-Rosenthal Theorem that
K
is completely complemented in any separable locally reflexive operator superspace,
K
the
C
∗
-algebra of compact operators on
ℓ
2
. M-cai's are also used in obtaining some special affirmative answers to the open problem of whether
K
is Banach-complemented in
A
for any separable
C
∗
-algebra
A
with
K⊂A⊂B(
ℓ
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 reflexive and X is
λ
-nuclear for some
λ≥1
; (iii) X is a closed 2-sided ideal in an operator algebra Y (via the Effros-Ruan result that then X has a contractive algebraic approximate identity). However it is shown that there exists a separable Banach space X which is an M-ideal in
Y=
X
∗∗
, yet X admits no M-approximate identity in Y.