On the growth of Hermitian groups

Palma, Rui

doi:10.4171/ggd/304

Cited by 5 publications

(1 citation statement)

References 7 publications

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“…where ν denotes the left Haar measure of G (This limit exists and it is a finite number greater or equal to 1; see [31,Theorem 1.5]). Then definite function (see [17]), and so, φ t,|•| ∈ 1≤p<∞ ℓ p (F d ).…”

Section: Integrable Haagerup Propertymentioning

confidence: 99%

Exotic C*-algebras of geometric groups

Samei¹,

Wiersma²

2018

Preprint

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We consider a new class of potentially exotic group C*-algebras C * (PF * p (G)) for a locally compact group G, and its connection with the class of potentially exotic group C*algebras C * L p (G) introduced by Brown and Guentner. Surprisingly, these two classes of C*algebras are intimately related. By exploiting this connection, we show C * L p (G) = C * (PF * p (G)) for p ∈ (2, ∞), and the C*-algebras C * L p (G) are pairwise distinct for p ∈ (2, ∞) when G belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of C * L p (G) and C * (PF * p (G)). This greatly generalizes earlier results of Okayasu (see [30]) and the second author (see [37]) on the pairwise distinctness of C * L p (G) for 2 < p < ∞ when G is either a noncommutative free group or the group SL(2, R), respectively.As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group G. Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem which presents sufficient condition implying the weak containment of a cyclic unitary representation of G in the left regular representation of G (see [14]). Also we give a near solution to a 1978 conjecture of Cowling stated in [10]. This conjecture of Cowling states if G is a Kunze-Stein group and π is a unitary representation of G with cyclic vector ξ such that the map G ∋ s → π(s)ξ, ξ belongs to L p (G) for some 2 < p < ∞, then Aπ ⊆ L p (G). We show Bπ ⊆ L p+ǫ (G) for every ǫ > 0 (recall Aπ ⊆ Bπ).

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Section: Integrable Haagerup Propertymentioning

confidence: 99%