A remark on trace properties of K-cycles

In this paper we discuss trace properties of $d^+$-summable $K$-cycles considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a trace theorem on the algebra \A of a $K$--cycle stated in [\rfr(Conn4)], namely we show that a natural functional on \A is a trace functional. Then we discuss whether this functional gives a trace on the whole universal graded differential algebra \Q(\A). On the one hand we prove that the regularity conditions on $K$-cycles considered in [\rfr(Conn4)] imply the trace property on \Q(\A). On the other hand, by constructing an explicit counterexample, we remark that the sole $K$-cycle assumption is not sufficient for such a property to hold.Comment: 11 pages, plain Te

Symmetries of L\'evy processes on compact quantum groups, their Markov semigroups and potential theory

Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced C${}^*$-algebra of a compact quantum group which are translation invariant (w.r.t. to the coproduct) are in one-to-one correspondence with L\'evy processes on its $*$-Hopf algebra. We use the theory of L\'evy processes on involutive bialgebras to characterize symmetry properties of the associated quantum Markov semigroup. It turns out that the quantum Markov semigroup is GNS-symmetric (resp. KMS-symmetric) if and only if the generating functional of the L\'evy process is invariant under the antipode (resp. the unitary antipode). Furthermore, we study L\'evy processes whose marginal states are invariant under the adjoint action. In particular, we give a complete description of generating functionals on the free orthogonal quantum group $O_n^+$ that are invariant under the adjoint action. Finally, some aspects of the potential theory are investigated. We describe how the Dirichlet form and a derivation can be recovered from a quantum Markov semigroup and its L\'evy process and we show how, under the assumption of GNS-symmetry and using the associated Sch\"urmann triple, this gives rise to spectral triples. We discuss in details how the above results apply to compact groups, group C$^*$-algebras of countable discrete groups, free orthogonal quantum groups $O_n^+$ and the twisted $SU_q (2)$ quantum group.Comment: 54 pages, thoroughly revised, to appear in the Journal of Functional Analysi

Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.Comment: Some proofs have been clarified, reference to previous literature is now more accurate, 33 pages, 6 figure

Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras

In this work we apply Noncommutative Potential Theory to prove (relative) amenability and the (relative) Haagerup Property $(H)$ of von Neumann algebras in terms of the spectral growth of Dirichlet forms. Examples deal with (inclusions of) countable discrete groups and free orthogonal compact quantum groups

Spectral triples for the Sierpinski Gasket

We construct a family of spectral triples for the Sierpinski Gasket $K$. For suitable values of the parameters, we determine the dimensional spectrum and recover the Hausdorff measure of $K$ in terms of the residue of the volume functional $a\to$ tr$(a\,|D|^{-s})$ at its abscissa of convergence $d_D$, which coincides with the Hausdorff dimension $d_H$ of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to the distance on $K$ induced by the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) $K$-theory is non-trivial. When the parameters belong to a suitable range, the abscissa of convergence $\delta_D$ of the energy functional $a\to$ tr$(|D|^{-s/2}|[D,a]|^2\,|D|^{-s/2})$ takes the value $d_E=\frac{\log(12/5)}{\log 2}$, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on $K$.Comment: 48 pages, 9 figures. Final version, to appear in J.Funct.Ana

Cost-effectiveness profile, organizational implications and patient preferences on the use of exogenous TSH therapy (Thyrogen®) vs. THW in thyroid residue ablation in Italy

BACKGROUND: Radioiodine ablation is an adjuvant procedure used to treat patients with differentiated thyroid cancer. For ablation to be successful, patients must have elevated levels of thyroid stimulating hormone (TSH). This can be achieved by withholding thyroid hormone therapy (endogenous stimulation), or by administration of recombinant human thyroid stimulating hormone (rhTSH; Thyrogen®; exogenous stimulation) to patients in the euthyroid state.AIM: To compare the estimated health benefits, cost and cost-effectiveness of TSH stimulation with and without Thyrogen® in the Italian setting.METHODS: A cost-utility analysis was undertaken to assess the impact of exogenous vs. endogenous TSH stimulation before radioiodine remnant ablation of patients with newly diagnosed, well-differentiated papillary or follicular thyroid cancer who have undergone total or near-total thyroidectomy. A Markov model was developed to simulate treatment costs and health outcomes associated with exogenous and endogenous stimulation in four distinct health states: pre-ablation, ablation, post-ablation, and well/recovery. Treatment was stratified by patients who receive high- and low-activity (30-100 mCi, respectively) in the ablation state. The Italian National Health System perspective was adopted in the base case scenario while the impact of indirect costs was explored in a sensitivity analysis. Costs and quality-adjusted life years (QALY) specific to each health state were estimated, summarized and converted into a corresponding incremental cost-effectiveness ratio (ICER).RESULTS: We calculated a cost-effectiveness ratio of 18,357.18 €/QALY gained whereas the inclusion of indirect cost and accident cost produced reductions of the ICER to € 14,609.51 and € 15,515.26 per QALY, respectively. Finally, all results in the sensitivity analysis are below the lower bound of national and international cost- effective threshold.CONCLUSION: Thyrogen® represents a cost-effective option for patients with differentiated thyroid cancer who underwent total or near-total thyroidectomy in Italy. Our findings are consistent with other cost-utility analyses