The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra B(E) of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c0 and ℓp for 1⩽p<∞. We add a new member to this family by showing that there are exactly four closed ideals in B(E) for the Banach space E≔(⊕ℓ2n)c0, that is, E is the c0-direct sum of the finite-dimensional Hilbert spaces ℓ21,ℓ22,…,ℓ2n,… .