Additivity of homological dimensions for a class of Banach algebras

Let Ω be a metrizable compact space. Suppose that its derived set of some finite order is empty. Let B be a unital Banach algebra, and let $\widehat \otimes $ stand for the projective tensor product. We prove the additivity formulas dg C(Ω)B $\widehat \otimes $ =dgB and db C(Ω) $\widehat \otimes $ B=dbC(Ω)+dbB for the global homological dimension and the homological bidimension. Thus these formulas are true for a new class of commutative Banach algebras in addition to those considered earlier by Selivanov.