Approximate identities for Ideals in L(L^p))

2024/02/21 by Johnson, William B., Schechtman, Gideon
#47L20 (Primary) 46E30 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2402.13438

Abstract

The main result is that the only non trivial closed ideal in the Banach algebra L(L^p) of bounded linear operators on L^p(0,1), 1≤ p < ∞, that has a left approximate identity is the ideal of compact operators. The algebra L(L¹) has at least one non trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.

Related

• Left approximate identities in algebras of compact operators on Banach spaces
• Ideals in L(L₁)
• One-sided ideals and approximate identities in operator algebras
• Ideal structure of the algebra of bounded operators acting on a Banach\n space
• Closed ideals with bounded Δ-weak approximate identities in some certain Banach algebras