Asymptotics of Matrix Coefficients and Closures of Fourier–Stieltjes Algebras

Mayer, Matthias

doi:10.1006/jfan.1996.2965

Cited by 19 publications

(18 citation statements)

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“…It is shown in [45,Thm. 4.5] that all connected groups for which (ε E , G E ) ∼ = (ε WAP , G WAP ) have Eberlein compactifications of the form illustrated in Proposition 4.1, below.…”

Section: Universal Property Of the Eberlein Compactificationmentioning

confidence: 96%

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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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“…It is shown in [45,Thm. 4.5] that all connected groups for which (ε E , G E ) ∼ = (ε WAP , G WAP ) have Eberlein compactifications of the form illustrated in Proposition 4.1, below.…”

Section: Universal Property Of the Eberlein Compactificationmentioning

confidence: 96%

“…Proof. For cases (a) and (b) it is shown in [9], and for case (c) it is shown in [45,Thm. 4.5], that WAP(G) ⊂ E(G).…”

Section: Universal Property Of the Eberlein Compactificationmentioning

confidence: 99%

Matrix coefficients of unitary representations and associated compactifications

Spronk¹,

Stokke²

2013

Indiana Univ. Math. J.

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“…This algebra is called the Fourier-Stieltjes algebra of G (see for example, [12,22]). The elements of this algebra will be called Fourier-Stieltjes functions on G.…”

Section: ) Note That If the Co-representationmentioning

confidence: 99%

“…Hence, always, B(G) ⊂ W AP (G). The question whether B(G) is dense in W AP (G) raised by Eberlein (see [31]) and leads to the following definition of the so-called Eberlein groups [22] (originally defined for locally compact groups). Definition 1.10.…”

Section: The Algebra B(g) Is Rarely Closed In C(g) Precisely If G Imentioning

confidence: 99%

Reflexively representable but not Hilbert representable compact flows and semitopological semigroups

Megrelishvili

2008

Colloq. Math.

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Abstract. We show that for many natural topological groups G (including the group Z of integers) there exist compact metric G-spaces (cascades for G = Z) which are reflexively but not Hilbert representable. This answers a question of T. Downarowicz. The proof is based on a classical example of W. Rudin and its generalizations. One more crucial step in the proof is our recent result which states that every weakly almost periodic function on a compact G-flow X comes from a G-representation of X on reflexive spaces. We also show that there exists a monothetic compact metrizable semitopological semigroup S which does not admit an embedding into the semitopological compact semigroup Θ(H) of all contractive linear operators for a Hilbert space H (though S admits an embedding into the compact semigroup Θ(V ) for certain reflexive V ).

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“…We call G totally minimal if any quotient by a closed normal subgroup G/S, admits no Hausdorff group topology which is strictly coarser that the quotient topology. In particular, for a totally minimal group locally compact group G we have that [29,30]) Let G be a connected locally compact group. Consider the condition (i) E(G ̟ ) is finite.…”

Section: Connected Groupsmentioning

confidence: 99%