Abstract. The analog of the Chern-Gauss-Bonnet theorem is studied for a C * -dynamical system consisting of a C * -algebra A equipped with an ergodic action of a compact Lie group G. The structure of the Lie algebra g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A ⊂ A as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique G-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on A and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C * -algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the "algebraic" existence of ergodic action and the "analytic" finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples -including noncommutative tori and quantum Heisenberg manifolds. Classification (2010). 46L87, 58B34. Mathematics Subject

Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results and Pimsner algebras. Under certain conditions, we prove that the fixed point algebra is purely infinite and nuclear. We further identify it as a Pimsner algebra, compute its K-theory and prove a "stability property": the fixed points only depend on the CQG via its fusion rules. We apply the theory to SU q (N ) and illustrate by explicit computations for SU q (2) and SU q (3). This construction provides examples of free actions of CQG (or "principal noncommutative bundles"). Subject Classification (2010). 46L80, 19K99, 81R50 Mathematics

We give a construction for lifting spectral triples to crossed products by Hilbert bimodules. The spectral triple one obtains is a concrete unbounded representative of the Kasparov product of the spectral triple and the Pimsner-Toeplitz extension associated to the crossed product by the Hilbert module. To prove that the lifted spectral triple is the above-mentioned Kasparov product, we rely on operator- *algebras and connexions.

In Section 1 an extensive bibliographical review is given for the classical and quantum virial and for the hypervirial theorems, with some references to the special and general relativistic cases. Classical mechanics concepts are discussed from the point of view of "objectivism." Some difficulties are examined concerning adiabatic and static approximations, partitioning, boundary 'conditions, constraints, and external interactions, and concepts used in analytical mechanics as related to the virial theorem. Connections of the quantum virial theorem to the Hellmann-Feynman theorem, force concept, partitioning and boundary conditions are mentioned briefly.In Section 2 the virial theorem is extended to periodic wave functions of the Bloch type. A corrective term, arising from the surface integral, takes account of the values of the wave function and its derivatives at the boundary of the integration space, values which are not null in the case of Bloch wave functions. The result is applied to solid state in the case of the monoelectronic approximation.