Magnus B. Landstad

Hecke C*-algebras, Schlichting completions and Morita equivalence

The Hecke algebra of a Hecke pair ( G , H ) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C *-completions of are addressed in terms of the projection using both Fells and Rieffels imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

A crossed-product approach to the Cuntz–Li algebras

Cuntz and Li have defined a C*-algebra associated to any integral domain, using generators and relations, and proved that it is simple and purely infinite and that it is stably isomorphic to a crossed product of a commutative C*-algebra. We give an approach to a class of C*-algebras containing those studied by Cuntz and Li, using the general theory of C*-dynamical systems associated to certain semidirect product groups. Even for the special case of the Cuntz-Li algebras, our development is new.

Hecke C*-algebras and semi-direct products

A Hecke pair (G,H) comprises a group G and a subgroup H for which every double coset is a finite union of left cosets, and the associated Hecke algebra, generated by the characteristic functions of double cosets, reduces to the group ∗-algebra of G/H when H is normal. In [4] we introduced the Schlichting completion as a tool for analyzing Hecke algebras, based in part upon work of Tzanev [13]. (A slight variation on this construction appears in [14].) The idea is that H is a compact open subgroup of G such that the Hecke algebra of (G,H) is naturally identified with the Hecke algebra H of (G,H). The characteristic function p of H is a projection in the group C∗-algebra A := C∗(G), and H can be identified with pCc(G)p ⊆ A; thus closure of H in A coincides with the corner pAp, which is Morita-Rieffel equivalent to the ideal ApA. In [4] we were mainly interested in studying when pAp is the enveloping C∗-algebra of the Hecke algebra H, and when the projection p is full in A, making the C∗-completion pAp ofHMorita-Rieffel equivalent to the group C∗-algebra A. We had most success when G = N o Q was a semidirect product with H contained in the normal subgroup N . In this paper we again consider G = N o Q, but now we allow the Hecke subgroup H to be spread across both N and Q. This leads to a refinement of the Morita-Rieffel equivalence ApA ∼ MR pAp (see

GENERALIZED HECKE ALGEBRAS AND C*-COMPLETIONS

For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra

QUANTIZATIONS ARISING FROM ABELIAN SUBGROUPS

\mathcal{H}_{\sigma}(G, H)

Compact and discrete subgroups of algebraic quantum groups I

, which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection

Exotic group C*-algebras in noncommutative duality

p_{\sigma} \in C_c(G_{\sigma}, \mathcal{B}(V_{\sigma}))

Groups with compact open subgroups and multiplier Hopf **-algebras

such that

Equivariant Deformations of Homogeneous Spaces

\mathcal{H}_{\sigma}(G, H)

Traces on Noncommutative Homogeneous Spaces

is isomorphic to