Approximate complementation and its applications in studying ideals of Banach algebras

Abstract

We show that a subspace of a Banach space having the approximation property inherits this property if and only if it is approximately complemented in the space. For an amenable Banach algebra a closed left, right or two-sided ideal admits a bounded right, left or two-sided approximate identity if and only if it is bounded approximately complemented in the algebra. If an amenable Banach algebra has a symmetric diagonal, then a closed left (right) ideal J has a right (resp. left) approximate identity (p α ) such that, for every compact subset K of J , the net (a · p α ) (resp. (p α · a)) converges to a uniformly for a ∈ K if and only if J is approximately complemented in the algebra.

Approximate diagonals and Følner conditions for amenable group and semigroup algebras

“…He shows that group algebras of compact groups always have approximate diagonals with this property [24,Proposition 6]. We note here that this is the only case in which a group algebra can have an approximate diagonal of this form.…”

“…The converse, which is proved in [24], is a special case of our Theorem 1.8: let m β (s, t) = e β (st) where (e β ) is a central bai for L 1 (G). Remarks 1.13.…”

“…The author would also like to thank Professors R. J. Loy and Y. Zhang for carefully reading and providing very useful comments upon earlier drafts of this paper. We further thank Professor Zhang for bringing the reference [24] to our attention. The financial support of the Natural Sciences and Engineering Research Council of Canada and the University of Alberta is gratefully acknowledged.…”

Approximately Biprojective Banach Algebras and Nilpotent Ideals

“…A very close and related notion to the (bounded) approximation property is (bounded) approximately complementation introduced and studied by Zhang in [17,18].…”

Projectivity of modules over Fourier algebras

“…As it is pointed out in Proposition 2.7, it is beneficial if we can determine whether a given algebra or a module has operator approximation property (OAP). In [28] and [29], Y. Zhang introduced the concept of being approximately complemented for a subspace E of a normed space X and showed that it is closely related to E having the approximation property. In this section, we modify this concept, so that it can be viewed in the category of operator spaces and we apply it to deduce OAP for certain Segal algebras.…”