Banach Journal of Mathematical Analysis (26622033)(3)
Let (X, d) be a compact metric space and A be a commutative and semisimple Banach algebra. Some of our recent works are related to the several BSE concepts of the vector-valued Lipschitz algebra Lip(X,A). In this paper as the main purpose, we verify the BED property for Lip(X,A), which is actually different from the BSE feature. We first prove as an elementary result that Lip(X,A) is regular if and only if A is so. Then we prove that A is a BED algebra, whenever Lip(X,A) is so. Afterwards, we verify the converse of this statement. Indeed, we prove that if A is a BED algebra then CBSE0(Δ(Lip(X,A)))⊆Lip(X,A)^ and LipX⊗A^⊆CBSE0(Δ(Lip(X,A))). It follows that if LipX⊗A is dense in Lip(X,A) then Lip(X,A) is a BED algebra, provided that A is so. Moreover, we conclude that the necessary and sufficient condition for the unital and in particular finite dimensional Banach algebra A, to be a BED algebra is that Lip(X,A) is a BED algebra. Finally, regarding to some known results which disapproves the BSE property for lipα(X,A)(0<α<1), we show that for any commutative and semisimple Banach algebra A with A0≠∅, lipα(X,A) fails to be a BED algebra, as well. © Tusi Mathematical Research Group (TMRG) 2024.
