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

A QHD-capable parallel H.264 decoder

Video coding follows the trend of demanding higher performance every new generation, and therefore could utilize many-cores. A complete parallelization of H.264, which is the most advanced video coding standard, was found to be difficult due to the complexity of the standard. In this paper a parallel implementation of a complete H.264 decoder is presented. Our parallelization strategy exploits function-level as well as data-level parallelism. Function-level parallelism is used to pipeline the H.264 decoding stages. Data-level parallelism is exploited within the two most time consuming stages, the entropy decoding stage and the macroblock decoding stage. The parallelization strategy has been implemented and optimized on three platforms with very different memory architectures, namely an 8-core SMP, a 64-core cc-NUMA, and an 18-core Cell platform. Evaluations have been performed using 4kx2k QHD sequences. On the SMP platform a maximum speedup of 4.5x is achieved. The SMP-implementation is reasonably performance portable as it achieves a speedup of 26.6x on the cc-NUMA system. However, to obtain the highest performance (speedup of 33.4x and throughput of 200 QHD frames per second), several cc-NUMA specific optimizations are necessary such as optimizing the page placement and statically assigning threads to cores. Finally, on the Cell platform a near ideal speedup of 16.5x is achieved by completely hiding the communication latency.EC/FP7/248647/EU/ENabling technologies for a programmable many-CORE/ENCOR

On the decomposition into Discrete, type II and type III C∗C^*-algebras

We obtained a "decomposition scheme" of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsido), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to "classify" C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking "essential extension" and "normal quotient". Furthermore, there exist the largest discrete finite ideal $A_{d,1}$, the largest discrete essentially infinite ideal $A_{d,\infty}$, the largest type II finite ideal $A_{II,1}$, the largest type II essentially infinite ideal $A_{II,\infty}$, and the largest type III ideal $A_{III}$ of any C*-algebra $A$ such that $A_{d,1} + A_{d,\infty} + A_{II,1} + A_{II,\infty} + A_{III}$ is an essential ideal of $A$. This "decomposition" extends the corresponding one for $W^*$-algebras. We also give a closer look at C*-algebras with Hausdorff primitive spectrum, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.Comment: 41 pages; we added a lot of details and some new result

Using OpenMP superscalar for parallelization of embedded and consumer applications

In the past years, research and industry have introduced several parallel programming models to simplify the development of parallel applications. A popular class among these models are task-based programming models which proclaim ease-of-use, portability, and high performance. A novel model in this class, OpenMP Superscalar, combines advanced features such as automated runtime dependency resolution, while maintaining simple pragma-based programming for C/C++. OpenMP Superscalar has proven to be effective in leveraging parallelism in HPC workloads. Embedded and consumer applications, however, are currently still mainly parallelized using traditional thread-based programming models. In this work, we investigate how effective OpenMP Superscalar is for embedded and consumer applications in terms of usability and performance. To determine the usability of OmpSs, we show in detail how to implement complex parallelization strategies such as ones used in parallel H.264 decoding. To evaluate the performance we created a collection of ten embedded and consumer benchmarks parallelized in both OmpSs and Pthreads.EC/FP7/248647/EU/ENabling technologies for a programmable many-CORE/ENCOR

Programming parallel embedded and consumer applications in OpenMP superscalar

In this paper, we evaluate the performance and usability of the parallel programming model OpenMP Superscalar (OmpSs), apply it to 10 different benchmarks and compare its performance with corresponding POSIX threads implementations