BED property for the tensor product of Banach algebras

Abstract

Let A and B be commutative and semisimple Banach algebras. Suppose that ∥ · ∥γ is an algebra cross-norm on A B such that ∥ · ∥γ ≥ ∥ · ∥e, and AbγB is a semisimple Banach algebra. In this paper, we verify the BED property for AbγB. In fact, we show that if AbγB is of BED, then both A and B are so, whenever either A or B is unital. We also show that if B (resp., A) is unital and Ab ⊆ CBSE0 (∆(A)) (resp., Bb ⊆ CBSE0 (∆(B))), then A\bγB ⊆ CBSE0 (∆(AbγB)). We also establish that if B (resp., A) is finite dimensional, then AbγB is of BED if and only if A (resp., B) is of BED. © 2024 Institute of Mathematics, University of Debrecen. All rights reserved.