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Homological Properties of Modules Over Group Algebras
Author(s) -
Dales H. G.,
Polyakov M. E.
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014686
Subject(s) - mathematics , injective function , group (periodic table) , banach algebra , pure mathematics , variety (cybernetics) , discrete group , mathematics subject classification , amenable group , algebra over a field , group algebra , property (philosophy) , discrete mathematics , banach space , philosophy , statistics , countable set , epistemology , chemistry , organic chemistry
Let G be a locally compact group, and let L 1 ( G ) be the Banach algebra which is the group algebra of G . We consider a variety of Banach left L 1 ( G )‐modules over L 1 ( G ), and seek to determine conditions on G that determine when these modules are either projective or injective or flat in the category. The answers typically involve G being compact or discrete or amenable. For example, in the case where G is discrete and 1 < p < ∞, we find that the module ℓ p ( G ) is injective whenever G is amenable, and that, if it is amenable, then G is ‘pseudo‐amenable’, a property very close to that of amenability. 2000 Mathematics Subject Classification 46H25, 43A20.