Abstract
We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.
Highlights
We study the classes of invariant and natural projections in the dual of a Banach algebra A
Why research invariant and natural projections? Invariant projections play a significant role concerning to the existence of bounded approximate identities in closed ideals of Banach algebras
Natural projections are related to the notion of weak bounded approximate identities ([5], p. 4164), i.e. given a closed ideal I of a Banach algebra A, a weak bounded approximate identity of I consists of a net {ui} ⊆ I so that ui, h → 1 for each h ∈ σ(I)
Summary
We study the classes of invariant and natural projections in the dual of a Banach algebra A. Let P(A∗) be the set of projections (or idempotents) of B(A∗). To the set of left, right and two sided invariant projections, respectively.
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