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On Minimal Ideals in Some Banach Algebras Associated with a Locally Compact Group
Author(s) -
Baker J. W.,
Filali M.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001733
Subject(s) - mathematics , pure mathematics , locally compact space , group (periodic table) , locally compact group , approximation property , algebra over a field , banach space , chemistry , organic chemistry
Let G be a non‐compact, locally compact group. The minimal ideals of the group algebra L 1 ( G ), the measure algebra M ( G ), and other Banach algebras (usually larger than L 1 ( G ) and M ( G )) such as the second dual, L 1 ( G )**, of L 1 ( G ) with an Arens product, or LUC( G )* with an Arens‐type product, are studied in the paper. Using integrable representations, which exist on some semisimple Lie groups, it is seen that the minimal left ideals can be of infinite dimension, and that the compactness of G is not necessary for these ideals to exist in L 1 ( G ) and M ( G ). It is shown also that, although the coefficients of integrable representations are minimal idempotents in LUC( G )* and L 1 ( G )** they do not generate minimal right ideals in these algebras, and that for a large class of groups, they do not generate minimal left ideals either.