West semigroups as compactifications of locally compact abelian groups

Elgun, Elcim

doi:10.1007/s00233-015-9747-8

Cited by 3 publications

(5 citation statements)

References 14 publications

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“…Example 4.9. We note some recent results of [16]. For G = Z, there is a representation π for which G π ∼ = L ∞ [0, 1] · ≤1 .…”

Section: 2mentioning

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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“…Example 4.9. We note some recent results of [16]. For G = Z, there is a representation π for which G π ∼ = L ∞ [0, 1] · ≤1 .…”

Section: 2mentioning

confidence: 96%

“…Example 2. 16. We note that if G is a group, the space of uniformly continuous bounded functions UCB(G) is * -closed where s * = s −1 for s in G. In general G U CB = G U CB(G) is not an involutive semigroup in our sense, i.e.…”

Section: 3mentioning

confidence: 99%

Matrix coefficients of unitary representations and associated compactifications

Spronk¹,

Stokke²

2013

Indiana Univ. Math. J.

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“…Thanks to Cowling and Rodway [5], B(G/S)| Z = B(Z), so E(G/S)| Z = E(Z). But E(Z) is infinite, thanks to Bouziad et al [3] (see, also, [10]). Hence, since any closed subsemigroup of a semitopological semigroup must contain an idempotent, we see that E((G/S) ̟ ), and hence E(G ̟ ), are each infinite.…”

Section: Connected Groupsmentioning

confidence: 95%

On operator amenability of Fourier-Stieltjes algebras

Spronk

2020

Bulletin des Sciences Mathématiques

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“…3 σ(A)-Eberlein functions for A is called BSE-functions (for A) in [23]. 4 We remark that a completely different concept with a similar name, that of the Eberlein algebra of a locally compact group G, is defined by Elgun [11] as the uniform closure of B (G).…”

Section: Preliminariesmentioning

confidence: 99%

“…In [33, Definition 2.2.9], Reiter and Stegeman call an algebra that satisfies the assumption in our Definition 2.9 in addition to some other conditions a Wiener algebra.3 σ(A)-Eberlein functions for A is called BSE-functions (for A) in[23] 4. We remark that a completely different concept with a similar name, that of the Eberlein algebra of a locally compact group G, is defined by Elgun[11] as the uniform closure of B (G).…”

mentioning

confidence: 99%

Characterisations of Fourier and Fourier–Stieltjes algebras on locally compact groups

Lau

Pham

2016

Advances in Mathematics

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Motivated by the beautiful work of M. A. Rieffel (1965) and of M. E. Walter (1974), we obtain characterisations of the Fourier algebra A(G) of a locally compact group G in terms of the class of F -algebras (i.e. a Banach algebra A such that its dual A ′ is a W * -algebra whose identity is multiplicative on A). For example, we show that the Fourier algebras are precisely those commutative semisimple F -algebras that are Tauberian, contain a nonzero real element, and possess a dual semigroup that acts transitively on their spectrums. Our characterisations fall into three flavours, where the first one will be the basis of the other two. The first flavour also implies a simple characterisation of when the predual of a Hopf-von Neumann algebra is the Fourier algebra of a locally compact group. We also obtain similar characterisations of the Fourier-Stieltjes algebras of G. En route, we prove some new results on the problem of when a subalgebra of A(G) is the whole algebra and on representations of discrete groups.2010 Mathematics Subject Classification. Primary 43A30, 22D25; Secondary 46J05, 22D10.

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West semigroups as compactifications of locally compact abelian groups

Cited by 3 publications

References 14 publications

Matrix coefficients of unitary representations and associated compactifications

Matrix coefficients of unitary representations and associated compactifications

On operator amenability of Fourier-Stieltjes algebras

Characterisations of Fourier and Fourier–Stieltjes algebras on locally compact groups

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