On the structure of the Fourier-Stieltjes algebra

Walter, Martin E.

doi:10.2140/pjm.1975.58.267

Cited by 18 publications

(20 citation statements)

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“…We introduce the concept of Eberlein containment and illustrate its relationship to more classical methods of comparing unitary representations. In particular we observe that the Eberlein compactification, which is the spectrum of the uniform closure of the Fourier-Stieltjes algebra in CB(G), is an invariant for G. In § §3.2 we study the spectra of algebras generated by matrix coefficients, giving a criterion for determining elements of the spectra of these algebras in the vein of Walter [64,63], and characterising theŠilov boundary within the spectrum. In § §3.3 we manipulate the role played by almost periodic functions to characterise amenability of G in terms of some uniformly closed algebras of functions.…”

Section: 2mentioning

confidence: 99%

Matrix coefficients of unitary representations and associated compactifications

Spronk¹,

Stokke²

2013

Indiana Univ. Math. J.

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We study, for a locally compact group G, the compactifications (π, G π ) associated with unitary representations π, which we call π-Eberlein compactifications. We also study the Gelfand spectra Φ A(π) of the uniformly closed algebras A(π) generated by matrix coefficients of such π. We note that Φ A(π) ∪ {0} is itself a semigroup and show that theŠilov boundary of A(π) is G π . We study containment relations of various uniformly closed algebras generated by matrix coefficients, and give a new characterisation of amenability: the constant function 1 can be uniformly approximated by matrix coefficients of representations weakly contained in the left regular representation if and only if G is amenable. We show that for the universal representation ω, the compactification (ω, G ω ) has a certain universality property: it is universal amongst all compactifications of G which may be embedded as contractions on a Hilbert space, a fact which was also recently proved by Megrelishvili [48]. We illustrate our results with examples including various abelian and compact groups, and the ax + b-group. In particular, we witness algebras A(π), for certain non-self-conjugate π, as being generalised algebras of analytic functions.

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Section: 2mentioning

confidence: 99%

Matrix coefficients of unitary representations and associated compactifications

Spronk¹,

Stokke²

2013

Indiana Univ. Math. J.

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“…It would be interesting to determine if this is true in general. It seems likely that completing the investigation begun in [41] might resolve this question.…”

Section: Theorem 412 Every Idempotent In B(h) Is An Element Of a * mentioning

confidence: 99%

The spine of a Fourier-Stieltjes algebra

Ilie

Spronk

2012

Proceedings of the London Mathematical Society

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Abstract. 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 nonquotient 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 * (G) contains the image of every completely bounded homomorphism from the Fourier algebra A(H) of any amenable group H. 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

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“…Similarly, e b ð1Þ (i.e., the functional defined by m) is the support projection of the trivial representation of G (this projection is denoted by z 0 in [Wal2], p. 271). To get a more concrete picture, one can show (in analogy to [Val], Th.…”

Section: Groups With the Kazhdan Property Tmentioning

confidence: 99%