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

Property (T) for non-unital C*-algebras

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C*-algebras. The stronger one of the two is called ``property (T)'' and the weaker one is called ``property (T_{e})''. It is shown that all non-unital C*-algebras do not have property (T) (neither do their unitalizations). Moreover, all non-unital $\sigma$-unital C*-algebras do not have property (T_e).Comment: 7 pages; to appear in J. Math. Anal. App

Effective Transfer of Pretrained Large Visual Model for Fabric Defect Segmentation via Specifc Knowledge Injection

Fabric defect segmentation is integral to textile quality control. Despite this, the scarcity of high-quality annotated data and the diversity of fabric defects present significant challenges to the application of deep learning in this field. These factors limit the generalization and segmentation performance of existing models, impeding their ability to handle the complexity of diverse fabric types and defects. To overcome these obstacles, this study introduces an innovative method to infuse specialized knowledge of fabric defects into the Segment Anything Model (SAM), a large-scale visual model. By introducing and training a unique set of fabric defect-related parameters, this approach seamlessly integrates domain-specific knowledge into SAM without the need for extensive modifications to the pre-existing model parameters. The revamped SAM model leverages generalized image understanding learned from large-scale natural image datasets while incorporating fabric defect-specific knowledge, ensuring its proficiency in fabric defect segmentation tasks. The experimental results reveal a significant improvement in the model's segmentation performance, attributable to this novel amalgamation of generic and fabric-specific knowledge. When benchmarking against popular existing segmentation models across three datasets, our proposed model demonstrates a substantial leap in performance. Its impressive results in cross-dataset comparisons and few-shot learning experiments further demonstrate its potential for practical applications in textile quality control.Comment: 13 pages,4 figures, 3 table

Cross-disciplinary collaboration through WuZhiQiao Project to foster cultural exchange and community engagement

In 2013, students of the Technological and Higher Education Institute of Hong Kong (THEi), with the support of WuZhiQiao (WZQ) Charitable Foundation, formed a core team of 11 students to organize and participate in social service projects to help the underprivileged in the Chinese mainland. WuZhiQiao (WZQ) projects, the first cross-region social service engagement by THEi students, bring together students from Hong Kong and the Mainland. WZQ Charitable Foundation aims to help the Chinese traditional village in building Pedestrian Bridge and organizing community projects. Since there are Chinese villages facing flooding during rainy seasons, the local villagers will be trapped inside the village without the chance to go outside or wade outside the village. There are hundreds of such villages and they highly need our help. Each project mainly involves two or three institutes from Hong Kong and the Mainland, and they organize the whole volunteer project including planning, investigation, design, promotion and operation. Through involvement in different states or provinces, WZQ projects provide good chance of communication and interaction between Hong Kong teams and the Mainland teams and advocate intercultural social services. The projects can foster the cultural exchange between Hong Kong and the Mainland. Moreover, the majority of WZQ project members are coming from the fields of engineering, architecture and health care. We can practice our learning from lectures through the project implementation. Different parties are involved in the engineering projects including clients, consultants, contractors, surveyors, engineers and workers. Engineering students can gain good understanding of the holistic picture of a real-life engineering project. We visited the location village for investigation to learn more about the local culture, geometry and the people’s needs and discussed with the Mainland Team through online chatting tools in order to propose the optimal pedestrian building design and other community projects. Having spent over six months in planning and preparation, THEi students will implement a bridge-building and community project in Chongqing in January 2015. Through engagement in this service-learning project, not only the undergraduates of THEi can benefit through personal development but the life quality of the disadvantaged can also be improved