Projective tensor product of protoquantum spaces

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a `matrix-free' language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for $L_1$ spaces. In particular, we obtain what could be called the proto-quantum version of the Grothendieck theorem about classical projective tensor products by $L_1$ spaces. At the end, we compare the new tensor product with the known projective tensor product of operator spaces, and show that the standard construction of the latter is not fit for general proto-quantum spaces

Fr\'echet Modules and Descent

We study several aspects of the study of Ind-Banach modules over Banach rings thereby synthesizing some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts.Comment: improved versio

Lexicographic cones and the ordered projective tensor product

We introduce lexicographic cones, a method of assigning an ordered vector space \Lex(S) to a poset $S$, generalising the standard lexicographic cone. These lexicographic cones are then used to prove that the projective tensor cone of two arbitrary cones is a cone, and to find a new characterisation of finite-dimensional vector lattices.Comment: 8 page

Entangled states close to the maximally mixed state

We give improved upper bounds on the radius of the largest ball of separable states of an m-qubit system around the maximally mixed state. The ratio between the upper bound and the best known lower bound (Hildebrand, quant.ph/0601201) thus shrinks to a constant c = \sqrt{34/27} ~ 1.122, as opposed to a term of order \sqrt{m\log m} for the best upper bound known previously (Aubrun and Szarek, quant.ph/0503221). We give concrete examples of separable states on the boundary to entanglement which realize these upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of four unit balls in R^3 and in the projective tensor product of an arbitrary number of unit balls in R^n for n = 2,4,8.Comment: 11 pages; v2: n qubit case adde

Operator space tensor products and the second dual of a Banach algebra.

This thesis explores a possible operator space framework for the study of the second dual of a Banach algebra A. We prove some new characterizations for A to be Arens regular and we try to unify, for the Arens regularity problem, two of current approaches: by considering weakly almost periodic functionals on A and by considering the topological center of A**. Motivated by this study, we define two operator space tensor products, namely, the extended projective tensor product and the normal projective tensor product. We investigate the properties of these two products, and compare them with other operator space tensor products. It is shown that the extended projective tensor product is injective, and the normal projective tensor product can linearize a class of bilinear maps under the condition that the pair of operator spaces has certain type of Kaplansky density property.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2006 .C366. Source: Masters Abstracts International, Volume: 45-01, page: 0332. Thesis (M.Sc.)--University of Windsor (Canada), 2006