On the index of product systems of Hilbert modules

In this note we prove that the set of all uniformly continuous units on a product system over a C* algebra B can be endowed with the structure of left right B - B Hilbert module after identifying similar units by the suitable equivalence relation. We use this construction to define the index of the initial product system, and prove that it is the generalization of earlier defined indices by Arveson (in the case B=C) and Skeide (in the case of spatial product system). We prove that such defined index is a covariant functor from the category od continuous product systems to the category of B bimodules. We also prove that the index is subadditive with respect to the outer tensor product of product systems, and prove additional properties of the index of product systems that can be embedded into a spatial one

TVS-cone metric spaces as a special case of metric spaces

There have been a number of generalizations of fixed point results to the so called TVS-cone metric spaces, based on a distance function that takes values in some cone with nonempty interior (solid cone) in some topological vector space. In this paper we prove that the TVS-cone metric space can be equipped with a family of mutually equivalent (usual) metrics such that the convergence (resp. property of being Cauchy sequence, contractivity condition) in TVS sense is equivalent to convergence (resp. property of being Cauchy sequence, contractivity condition) in all of these metrics. As a consequence, we prove that if a topological vector space $E$ and a solid cone $P$ are given, then the category of TVS-cone metric spaces is a proper subcategory of metric spaces with a family of mutually equivalent metrics (Corollary 3.9). Hence, generalization of a result from metric spaces to TVS-cone metric spaces is meaningless. This, also, leads to a formal deriving of fixed point results from metric spaces to TVS-cone metric spaces and makes some earlier results vague. We also give a new common fixed point result in (usual) metric spaces context, and show that it can be reformulated to TVS-cone metric spaces context very easy, despite of the fact that formal (syntactic) generalization is impossible. Apart of main results, we prove that the existence of a solid cone ensures that the initial topology is Hausdorff, as well as it admits a plenty of convex open sets. In fact such topology is stronger then some norm topology.Comment: 14 page

On nonlinear quasi-contractions on TVS-cone metric spaces

AbstractRecently, Du [W.-S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. (2009), doi:10.1016/j.na.2009.10.026] introduced the notion of TVS-cone metric space. In this paper we present fixed point theorem for nonlinear quasi-contractive mappings defined on TVS-cone metric space, which generalizes earlier results obtained by Ilić and Rakočević [D. Ilić, V. Rakočević, Quasi-contractions on a cone metric space, Appl. Math. Lett. 22 (2009) 728–731] and Kadelburg, Radenović and Rakočević [Z. Kadelburg, S. Radenović, V. Rakočević, Remarks on quasi-contractions on a cone metric space, Appl. Math. Lett. 22 (2009) 1674–1679]