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.

ABSTRACT. We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1.We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A u (F ) is not of Lie type. We discuss an example arising from the compact real form of U q (sl 2 ) for q < 0.

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies * -regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C * -norm.

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We study quantum chains whose Hamiltonians are perturbations by interactions of short range of a Hamiltonian consisting of a sum of on-site terms that do not couple the degrees of freedom located at different sites of the chain and have a strictly positive energy gap above their ground-state energy. For interactions that are form-bounded w.r.t. the on-site terms, we prove that the spectral gap of the perturbed Hamiltonian above its ground-state energy is bounded from below by a positive constant uniformly in the length of the chain, for small values of a coupling constant. Our proof is based on an extension of a novel method introduced in [FP] involving local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the chain.