Reduced K-theory for Azumaya algebras

Abstract In the theory of central simple algebras, often we are dealing with abelian groups which arise from the kernel or co-kernel of functors which respect transfer maps (for example K-functors). Since a central simple algebra splits and the functors above are “trivial” in the split case, one can prove certain calculus on these functors. The common examples are kernel or co-kernel of the maps Ki(F)?Ki(D), where Ki are Quillen K-groups, D is a division algebra and F its center, or the homotopy fiber arising from the long exact sequence of above map, or the reduced Whitehead group SK1. In this note we introduce an abstract functor over the category of Azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it. This, for example, immediately shows that K-theory of an Azumaya algebra over a local ring is “almost” the same as K-theory of the base ring. The main result is to prove that reduced K-theory of an Azumaya algebra over a Henselian ring coincides with reduced K-theory of its residue central simple algebra. The note ends with some calculation trying to determine the homotopy fibers mentioned above.

On subspace lattices II. Continuity of Lat

We study the continuity of the map Lat sending an ultraweakly closed operator algebra to its invariant subspace lattice. We provide an example showing that Lat is in general discontinuous and give sufficient conditions for the restricted continuity of this map. As consequences we obtain that Lat is continuous on the classes of von Neumann and Arveson algebras and give a general approximative criterion for reflexivity, which extends Arvesonâ??s theorem on the reflexivity of commutative subspace lattices.

On subspace lattices I. Closedness type properties and tensor products

We study properties of subspace lattices related to the continuity of the map Lat and the notion of reflexivity. We characterize various “closedness” properties in different ways and give the hierarchy between them. We investigate several properties related to tensor products of subspace lattices and show that the tensor product of the projection lattices of two von Neumann algebras, one of which is injective, is reflexive.

A Novel Multimodal Interface for Improving Visually Impaired People's Web Accessibility

This paper introduces a novel interface designed to help blind and visually impaired people to explore and navigate on the Web. In contrast to traditionally used assistive tools, such as screen readers and magnifiers, the new interface employs a combination of both audio and haptic features to provide spatial and navigational information to users. The haptic features are presented via a low-cost force feedback mouse allowing blind people to interact with the Web, in a similar fashion to their sighted counterparts. The audio provides navigational and textual information through the use of non-speech sounds and synthesised speech. Interacting with the multimodal interface offers a novel experience to target users, especially to those with total blindness. A series of experiments have been conducted to ascertain the usability of the interface and compare its performance to that of a traditional screen reader. Results have shown the advantages that the new multimodal interface offers blind and visually impaired people. This includes the enhanced perception of the spatial layout of Web pages, and navigation towards elements on a page. Certain issues regarding the design of the haptic and audio features raised in the evaluation are discussed and presented in terms of recommendations for future work.