Property (T)(T) and strong Property (T)(T) for unital C∗C^*-algebras

In this paper, we will give a thorough study of the notion of Property $(T)$ for $C^*$-algebras (as introduced by M.B. Bekka in \cite{Bek-T}) as well as a slight stronger version of it, called "strong property $(T)$" (which is also an analogue of the corresponding concept in the case of discrete groups and type $\rm II_1$-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property $(T)$ and strong property $(T)$. We will also relate them to certain $(T)$-type properties of the unitary group of the underlying $C^*$-algebra

The Growth of Dental Implant Literature from 1966 to 2016: A Bibliometric Analysis

This bibliometric book chapter overviewed the dental implant literature from 1966 to 2016 via the Web of Science database. Articles and reviews published by 2016 on the topic of dental implants were identified and analyzed in terms of their authors, affiliations, countries/territories of the affiliations, journal title and journal category. The performance indices of the 10 journals with the highest numbers of dental implant publications were extracted from Journal Citation Reports. A total of 14,335 articles or reviews were published in 1081 academic journals, with majority (10,487; 73.2%) in dental journals. With 317,263 total citations, each publication was cited 22.1 times on average. About 10 journals accounted for 47.0% of total publications, five dedicated to dental implants. Performance indices of journals publishing dental implant manuscripts have been stable over the last decade. Clinical Oral Implants Research was the best performing journal among them in 2016

Linear orthogonality preservers of Hilbert bundles

Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $\theta$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if \left =0 whenever \left =0. We prove that if $\theta$ is an orthogonality preserving map from a full Hilbert $C_0(\Omega)$-module $E$ into another Hilbert $C_0(\Omega)$-module $F$ that satisfies a weaker notion of $C_0(\Omega)$-linearity (known as "localness"), then $\theta$ is bounded and there exists $\phi\in C_b(\Omega)_+$ such that \left\ =\ \phi\cdot\left, \quad \forall x,y \in E. On the other hand, if $F$ is a full Hilbert $C^*$-module over another commutative $C^*$-algebra $C_0(\Delta)$, we show that a "bi-orthogonality preserving" bijective map $\theta$ with some "local-type property" will be bounded and satisfy \left\ =\ \phi\cdot\left\circ\sigma, \quad \forall x,y \in E where $\phi\in C_b(\Omega)_+$ and $\sigma: \Delta \rightarrow \Omega$ is a homeomorphism

Linear orthogonality preservers of Hilbert C∗C^*-modules over general C∗C^*-algebras

As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $\Phi : E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$\langle \Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0,$$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$\langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E).$$ As a consequence, $\Phi$ is automatically bounded, the induced map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$.Comment: 15 page

Role of the Hypoxia-Inducible Factor in Periodontal Inflammation

Human periodontitis is a chronic inflammatory disease induced by opportunistic Gram-negative anaerobic bacteria at the tooth-supporting apparatus. Within the gingivitis-affected sulcus or periodontal pocket, the resident anaerobic bacteria interact with the host inflammatory reactions leading to a lower oxygen or hypoxic environment. A cellular/tissue oxygen-sensing mechanism and its appropriate regulation are needed to assist tissue adaptation to natural/pathology-induced variations in oxygen availability. In this chapter, we reviewed the biological relevance of hypoxia in periodontal/oral cellular development, epithelial barrier function, periodontal inflammation, and immunity. The role of hypoxia-inducible factor-1α in pathogen-host cross talk and alveolar bone homeostasis was also discussed. The naturally occurring pathophysiological process of hypoxia appeared to entail fundamental relevance for periodontal defense and regeneration