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TY - JOUR
T1 - Finitely-generated left ideals in Banach algebras on groups and semigroups
AU - White, Jared
PY - 2017/4/18
Y1 - 2017/4/18
N2 - Let G be a locally compact group. We prove that the augmentation ideal in L1(G) is (algebraically) finitely-generated as a left ideal if and only if G is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason.
AB - Let G be a locally compact group. We prove that the augmentation ideal in L1(G) is (algebraically) finitely-generated as a left ideal if and only if G is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use the theory that we have developed to construct some examples of commutative Banach algebras that relate to a theorem of Gleason.
U2 - 10.4064/sm8743-1-2017
DO - 10.4064/sm8743-1-2017
M3 - Journal article
VL - 239
SP - 67
EP - 99
JO - Studia Mathematica
JF - Studia Mathematica
SN - 0039-3223
ER -