Normed “upper interval” algebras without nontrivial closed subalgebras

C. J. Read¹

¹ Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England

Studia Mathematica, Tome 171 (2005) no. 3, pp. 295-303

Résumé

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.

Détail

DOI : 10.4064/sm171-3-6

Keywords: long standing problem whether there infinite dimensional commutative banach algebra without nontrivial closed ideals sense banach algebraists counterpart invariant subspace problem banach spaces here solve famous problem solve related problem finding necessarily commutative infinite dimensional normed algebras which even have nontrivial closed subalgebras examples incomplete normed algebras rather banach algebras problem finding algebras posed elazko until only normed algebras general topological algebras construction here kept short because uses key lemma involved construction lrrw algebra loy read runde willis algebras dense subalgebras certain commutative banach algebras compact multiplication