BED property for the tensor product of Banach algebras

Let A andB be commutative and semisimple Banach algebras. Supposethat parallel to<middle dot>parallel to gamma is an algebra cross-norm onA circle times Bsuch that parallel to<middle dot>parallel to gamma >= parallel to<middle dot>parallel to e, andAb circle times gamma Bisa semisimple Banach algebra. In this paper, we verify the BED property forAb circle times gamma B.In fact, we show that ifAb circle times gamma Bis of BED, then bothAandBare so, whenever eitherAorBis unital. We also show that ifB(resp.,A) is unital andbA subset of C0BSE(triangle(A))(resp.,bB subset of C0BSE(triangle(B))), then\Ab circle times gamma B subset of C0BSE(triangle(Ab circle times gamma B)). We also establish that ifB(resp.,A) is finite dimensional, thenAb circle times gamma Bis of BED if and only ifA(resp.,B) is of BED.