We initiate a detailed and systematic study of automorphisms of the Cuntz algebras O n which preserve both the diagonal and the core U HF -subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of Aut(O n ) and leads to numerous new examples. In particular, we completely classify all such automorphisms of O 2 for the permutation unitaries in ⊗ 4 M 2 . We show that the subgroup of Out(O 2 ) generated by these automorphisms contains a copy of the infinite dihedral group Z ⋊ Z 2 .

We consider the class of``localized endomorphisms'' of the Cuntz algebras and we make some computations on the index of the associated endomorphisms of type III * factors.

An inclusion of observable nets satisfying duality induces an inclusion of canonical field nets. Any Bose net intermediate between the observable net and the field net and satisfying duality is the fixed-point net of the field net under a compact group. This compact group is its canonical gauge group if the occurrence of sectors with infinite statistics can be ruled out for the observable net and its vacuum Hilbert space is separable.Comment: 28 pages, LaTe

Let F be a local net of von Neumann algebras in four spacetime dimensions satisfying certain natural structural assumptions. We prove that if F has trivial superselection structure then every covariant, Haag-dual subsystem B is of the form F G 1 ⊗ I for a suitable decomposition F = F 1 ⊗ F 2 and a compact group action. Then we discuss some application of our result, including free field models and certain theories with at most countably many sectors.

This paper is an invitation to Fourier analysis in the context of reduced twisted C*-crossed products associated with discrete unital twisted C*-dynamical systems. We discuss norm-convergence of Fourier series, multipliers and summation processes. Our study relies in an essential way on the (covariant and equivariant) representation theory of C * -dynamical systems on Hilbert C*-modules. It also yields some information on the ideal structure of reduced twisted C*-crossed products.

We undertake a systematic study of the so-called 2-adic ring C * -algebra Q 2 . This is the universal C * -algebra generated by a unitary U and an isometry S 2 such that S 2 U = U 2 S 2 and S 2 S * 2 + U S 2 S * 2 U * = 1. Notably, it contains a copy of the Cuntz algebrathrough the injective homomorphism mapping S 1 to U S 2 . Among the main results, the relative commutant C * (S 2 ) ′ ∩ Q 2 is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion O 2 ⊂ Q 2 , namely the endomorphisms of Q 2 that restrict to the identity on O 2 are actually the identity on the whole Q 2 . Moreover, there is no conditional expectation from Q 2 onto O 2 . As for the inner structure of Q 2 , the diagonal subalgebra D 2 and C * (U ) are both proved to be maximal abelian in Q 2 . The maximality of the latter allows a thorough investigation of several classes of endomorphisms and automorphisms of Q 2 . In particular, the semigroup of the endomorphisms fixing U turns out to be a maximal abelian subgroup of Aut(Q 2 ) topologically isomorphic with C(T, T). Finally, it is shown by an explicit construction that Out(Q 2 ) is uncountable and non-abelian.

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C * -algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.MSC 1991: 22D10, 22D25, 46L55, 43A07, 43A65

Abstract. We introduce and study several notions of amenability for unitary corepresentations and * -representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C * -categories.