Spaces of compact operators on 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 2 m ⊕ [ 0 , α ] , the topological sum of Cantor cubes 2 m and the intervals of ordinal numbers [ 0 , α ] . More precisely, we prove that if 2 m and ℵ γ are not real-valued measurable cardinals and n ⩾ ℵ 0 is not sequential cardinal, then for every ordinals ξ, η, λ and μ with ξ ⩾ ω 1 , η ⩾ ω 1 , λ = μ < ω or λ , μ ∈ [ ω γ , ω γ + 1 [ , the following statements are equivalent:(a) ( 2 m ⊕ [ 0 , λ ] ) , C ( 2 n ⊕ [ 0 , ξ ] ) ) and K ( C ( 2 m ⊕ [ 0 , μ ] ) , C ( 2 n ⊕ [ 0 , η ] ) ) are isomorphic. C ( [ 0 , ξ ] ) is isomorphic to C ( [ 0 , η ] ) or C ( [ 0 , ξ ] ) is isomorphic to C ( [ 0 , α p ] ) and C ( [ 0 , η ] ) is isomorphic to C ( [ 0 , α q ] ) for some regular cardinal α and finite ordinals p ≠ q . it is relatively consistent with ZFC that this result furnishes a complete isomorphic classification of these spaces of compact operators.