ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 22, 46 Bergmann, Wolfgang Reinhold, Università della Basilicata, Potenza, Italy Amenable Groups and Generalized Cuntz Algebras Consider the C^{∗}-category C with arrow sets (r, s) := B(⊗^{r}H, ⊗^{s}H), H Hilbert space. Identifying (r, s) inside (r + 1, s + 1) with T ↦ T ⊗ I starts a process which collapses C into O_{H}, the graded generalized Cuntz C^{∗}- algebra over H, obtained by completion from the (suitably normed) *- algebra ∑_{k∈ZZ} lim_{→,r} (r, r + k). The action of V ∈ U(H) on the arrows of C given by T ∈ (r, s) ↦ ⊗^{s}V ∘ T ∘ ⊗^{r}V^{∗} ∈ (r, s) resists this construction, giving the action Λ of U(H) on O_{H}. Results: We introduce, by infinite tensor product and Rieffel induction, a natural family S of representations of O_{H}. With G noncompact Lie group and λ the left regular representation of G on H = L^{2}(G), we get: G is amenable IFF there exist infinite tensor powers of λ IFF there exist, in S, irreducible Λ∘λ-covariant representations of O_{H}. We started from a conjecture of S. Doplicher. Reference: M. Aita, W. R. Bergmann and R. Conti, Amenable Groups and Generalized Cuntz Algebras, J. Funct. Anal. 150 (1997), 48–64.