Exotic C⁎-algebras of geometric groups

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 [40] ) 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 A π ⊆ L p ( G ) . We show B π ⊆ L p + ϵ ( G ) for every ϵ > 0 (recall A π ⊆ B π ).

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.