Proceedings of the London Mathematical Society Advance Access originally published online on November 27, 2006
Proceedings of the London Mathematical Society 2007 94(2):273-301; doi:10.1112/plms/pdl002
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© 2006 London Mathematical Society
The spine of a Fourier-Stieltjes algebra
1 Department of Mathematical Sciences
Lakehead University
955 Oliver Road
Thunder Bay, ON P7B 5E1
Canada
milie{at}lakeheadu.ca
2 Department of Pure Mathematics
University of Waterloo
Waterloo, ON N2L 3G1
Canada
nspronk{at}uwaterloo.ca
Received 21 July 2005. Revision received 29 March 2006.
We define the spine A *(G) of the Fourier–Stieltjes algebra B (G) of a locally compact group G. This algebra encodes information about much of the fine structure of B (G), particularly information about certain homomorphisms and idempotents.
We show that A *(G) is graded over a certain semi-lattice, that of non-quotient locally precompact topologies on G. We compute the spine's spectrum G*, which admits a semi-group structure. We discuss homomorphisms from A *(G) to B (H) where H is another locally compact group; and we show that A *(H) contains the image of every completely bounded homomorphism from the Fourier algebra A (H) of any amenable group G. We also show that A *(G) contains all of the idempotents in B (G). Finally, we compute examples for vector groups, abelian lattices, minimally almost periodic groups and the (ax + b)-group; and we explore the complexity of A *(G) for the discrete rational numbers and free groups.