Additivity of homological dimensions for tensor products of some Banach algebras

It is proved that if A = C(Ω), where Ω is an infinite metrizable compact space such that some finite-order iterated derived set of Ω is empty, then for every unital Banach algebra B the global dimensions and the bidimensions of the Banach algebras A \(\hat \otimes \) B and B are related as dg A \(\hat \otimes \) B = 2 + dg B and db A \(\hat \otimes \) B = 2 + db B. Thus, a partial extension of Selivanov’s result is obtained.