### A canonical trace associated with certain spectral triples.

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In this paper we aim for a generalization of the Steenrod Approximation Theorem from [16, Section 6.7], concerning a smoothing procedure for sections in smooth locally trivial bundles. The generalization is that we consider locally trivial smooth bundles with a possibly infinite-dimensional typical fibre. The main result states that a continuous section in a smooth locally trivial bundles can always be smoothed out in a very controlled way (in terms of the graph topology on spaces of continuous...

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...

Let $\Omega $ be a bounded open subset of ${\mathbb{R}}^{n}$, $n>2$. In $\Omega $ we deduce the global differentiability result $$u\in {H}^{2}(\Omega ,{\mathbb{R}}^{N})$$ for the solutions $u\in {H}^{1}(\Omega ,{\mathbb{R}}^{n})$ of the Dirichlet problem $$u-g\in {H}_{0}^{1}(\Omega ,{\mathbb{R}}^{N}),-\sum _{i}{D}_{i}{a}^{i}(x,u,Du)={B}_{0}(x,u,Du)$$ with controlled growth and nonlinearity $q=2$. The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

We propose a new construction of characteristic classes for noncommutative algebraic principal bundles (Hopf-Galois extensions) with values in Hochschild and cyclic homology.

S. L. Woronowicz's theory of C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions...

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${\mathcal{M}}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator ${L}_{\xi}$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)={e}^{i{L}_{\xi}}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=({L}_{\xi}-i){({L}_{\xi}+i)}^{-1}$. We also show that the unitary group ${U}_{\mathcal{M}}\subset L\xb2(\mathcal{M},\tau )$ with the strong operator topology is not an embedded submanifold...