Normed “upper interval” algebras without nontrivial closed subalgebras
It is a long standing open problem whether there is any infinite-dimensional commutative Banach algebra without nontrivial closed ideals. This is in some sense the Banach algebraists' counterpart to the invariant subspace problem for Banach spaces. We do not here solve this famous problem, but solve a related problem, that of finding (necessarily commutative) infinite-dimensional normed algebras which do not even have nontrivial closed subalgebras. Our examples are incomplete normed algebras rather than Banach algebras. The problem of finding such algebras, posed by W. Żelazko, was until now open not only for normed algebras but for more general topological algebras. Our construction here is kept short because it uses a key lemma involved in the construction of the “LRRW algebra” of Loy, Read, Runde and Willis. The algebras we find are dense subalgebras of certain commutative Banach algebras with compact multiplication.