Invariant subspaces of some function spaces on a locally compact group |
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Authors: | Rodney Nillsen |
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Institution: | Department of Mathematics, The University of Wollongong, Wollongong, New South Wales, 2500, Australia |
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Abstract: | Let G be a σ-compact and locally compact group. If f?L∞(G) let Uf be the closed subspace of L∞(G) generated by the left translations of f. Conditions are given which ensure that each function in Uf may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case Uf contains subspaces which are isometrically isomorphic to l1. If G is metrizable and nondiscrete there is a continuum Γ in L∞(G) such that, for each f?Γ, Uf contains no non-zero continuous function, and for f, g?Γ with f ≠ g, Uf ∩ Ug = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, Uf contains no non-zero left uniformly continuous function, and for f, g?Γ with f ≠ g, Uf ∩ Ug = {0}. The subspaces Uf above are translation invariant but are not convolution invariant. |
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