Normed "upper interval" algebras without nontrivial closed subalgebras

C. J. Read. "Normed "upper interval" algebras without nontrivial closed subalgebras." Studia Mathematica 171.3 (2005): 295-303. <http://eudml.org/doc/284486>.

@article{C2005,
abstract = {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.},
author = {C. J. Read},
journal = {Studia Mathematica},
keywords = {closed ideal problem; non-trivial closed subalgebra; Banach algebra},
language = {eng},
number = {3},
pages = {295-303},
title = {Normed "upper interval" algebras without nontrivial closed subalgebras},
url = {http://eudml.org/doc/284486},
volume = {171},
year = {2005},
}

TY - JOUR
AU - C. J. Read
TI - Normed "upper interval" algebras without nontrivial closed subalgebras
JO - Studia Mathematica
PY - 2005
VL - 171
IS - 3
SP - 295
EP - 303
AB - 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.
LA - eng
KW - closed ideal problem; non-trivial closed subalgebra; Banach algebra
UR - http://eudml.org/doc/284486
ER -