Vol. 141, No. 2, 1990

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Complementation of certain subspaces of Lāˆž(G) of a locally compact group

Anthony To-Ming Lau and Viktor Losert

Vol. 141 (1990), No. 2, 295ā€“310
Abstract

Let G be a locally compact group, WAP(G) be the space of continuous weakly almost periodic functions on G and C0(G) the space of continuous functions on G vanishing at infinity. We prove in this paper, among other things, that if G is infinite and X is any subspace of WAP(G) (or CB(G), the space of bounded continuous functions in case G is nondiscrete) containing C0(G), then X is uncomplemented in L(G). If G is non-compact, then WAP(G) is uncomplemented in LUC(G). Furthermore, AP(G), the space of continuous almost periodic functions on G, is complemented in LUC(G) if and only if G∕N is compact, where N is the intersection of the kernels of all finite-dimensional continuous unitary representations of G. We also prove that if A is any left translation invariant C-subalgebra of C0(G), then A is the range of a continuous projection commuting with left translations.

Mathematical Subject Classification 2000
Primary: 43A60
Secondary: 22D15
Milestones
Received: 29 September 1987
Revised: 18 September 1988
Published: 1 February 1990
Authors
Anthony To-Ming Lau
http://www.math.ualberta.ca/Lau_A.html
Viktor Losert