Spaces of compact operators on C(m2⊕[0,α]) spaces

We classify up to isomorphism the spaces of compact operators K(E,F), where E and F are Banach spaces of all continuous functions defined on the compact spaces m2⊕[0,α], the topological sum of Cantor cubes m2 and the intervals of ordinal numbers [0,α]. More precisely, we prove that if m2 and ℵγ are not real-valued measurable cardinals and n⩾ℵ0 is not sequential cardinal, then for every ordinals ξ, η, λ and μ with ξ⩾ω1, η⩾ω1, λ=μ<ω or λ,μ∈[ωγ,ωγ+1[, the following statements are equivalent:(a)K(C(m2⊕[0,λ]),C(n2⊕[0,ξ])) and K(C(m2⊕[0,μ]),C(n2⊕[0,η])) are isomorphic.(b)Either C([0,ξ]) is isomorphic to C([0,η]) or C([0,ξ]) is isomorphic to C([0,αp]) and C([0,η]) is isomorphic to C([0,αq]) for some regular cardinal α and finite ordinals p≠q. Thus, it is relatively consistent with ZFC that this result furnishes a complete isomorphic classification of these spaces of compact operators.