Bounded and unitary elements in pro-C^*-algebras

A pro-C^*-algebra is a (projective) limit of C^*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C^*-algebras can be seen as non-commutative k-spaces. An element of a pro-C^*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C^*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C^*-algebra. In this paper, we investigate pro-C^*-algebras from a categorical point of view. We study the functor (-)_b that assigns to a pro-C^*-algebra the C^*-algebra of its bounded elements, which is the dual of the Stone-\v{C}ech-compactification. We show that (-)_b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand-duality for commutative unital pro-C^*-algebras is also presented.Comment: v2 (accepted

SUPPA2:fast, accurate, and uncertainty-aware differential splicing analysis across multiple conditions

Supplementary methods. (PDF 315 kb

Magyar Tanítóképző 57 (1944) 4

Magyar Tanítóképző A Tanítóképző-intézeti Tanárok Országos Egyesületének folyóirata 57. évfolyam, 4. szám Budapest, 1944. áprili

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

CHD8 suppression impacts on histone H3 lysine 36 trimethylation and alters RNA alternative splicing

Disruptive mutations in the chromodomain helicase DNA-binding protein 8 gene (CHD8) have been recurrently associated with autism spectrum disorders (ASDs). Here we investigated how chromatin reacts to CHD8 suppression by analyzing a panel of histone modifications in induced pluripotent stem cell-derived neural progenitors. CHD8 suppression led to significant reduction (47.82%) in histone H3K36me3 peaks at gene bodies, particularly impacting on transcriptional elongation chromatin states. H3K36me3 reduction specifically affects highly expressed, CHD8-bound genes and correlates with altered alternative splicing patterns of 462 genes implicated in ‘regulation of RNA splicing’ and ‘mRNA catabolic process’. Mass spectrometry analysis uncovered a novel interaction between CHD8 and the splicing regulator heterogeneous nuclear ribonucleoprotein L (hnRNPL), providing the first mechanistic insights to explain the CHD8 suppression-derived splicing phenotype, partly implicating SETD2, a H3K36me3 methyltransferase. In summary, our results point toward broad molecular consequences of CHD8 suppression, entailing altered histone deposition/maintenance and RNA processing regulation as important regulatory processes in ASD