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

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

Ready for 21st-century Education – Pre-service Music Teachers Embracing ICT to Foster Student-centered Learning

AbstractThere is always competition for curriculum time for pre-service teachers, especially in teaching methods courses. This study explored whether pre-service teachers are prepared to learn from each other online. It looked at 47 students taking a course entitled “Music Education Methods and Strategies”. Each was required to design a game as a form of pedagogy in music education and to upload it to a learning management system (LMS). They then had to read and comment on all submissions and vote for the best three games. After doing so, the students completed an online survey about their experiences. Thirty-nine valid questionnaires were submitted, giving a response rate of 83%. It is very encouraging to note that the students gave all questions a high rating (scoring them 3 or above on a 5-point Likert-type scale). The two top-rated items confirmed that the students felt responsible for their own learning and competent in using a LMS. This suggests that pre-service music teachers are ready for 21st-century education using information and communications technology to develop learner-centered activities