Martino Lupini

High density piecewise syndeticity of sumsets

Abstract Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A + B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B , there is a high density set of witnesses to the piecewise syndeticity of A + B . Most of the results are shown to hold more generally for subsets of Z d . The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.

Model-theoretic aspects of the Gurarij operator system

We establish some of the basic model theoretic facts about the Gurarij operator system GS recently constructed by the second-named author. In particular, we show: (1) GS is the unique separable 1-exact existentially closed operator system; (2) GS is the unique separable nuclear model of its theory; (3) every embedding of GS into its ultrapower is elementary; (4) GS is the prime model of its theory; and (5) GS does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any q ∈ ℕ, the theories of Mq-spaces and Mq-systems do have a model companion, namely the Fra¨ıssé limit of the class of finite-dimensional Mq-spaces and Mq-systems respectively; moreover, we show that the model companion is separably categorical. We conclude the paper by showing that no C* algebra can be existentially closed as an operator system.

Operator space and operator system analogs of Kirchberg's nuclear embedding theorem

Abstract The Gurarij operator space NG introduced by Oikhberg is the unique separable 1-exact operator space that is approximately injective in the category of 1-exact operator spaces and completely isometric linear maps. We prove that a separable operator space X is nuclear if and only if there exist a linear complete isometry φ : X → NG and a completely contractive projection from NG onto the range of φ. This can be seen as the operator space analog of Kirchbergs nuclear embedding theorem. With similar methods we also establish the natural operator system analog of Kirchbergs nuclear embedding theorem involving the Gurarij operator system GS .

An Intrinsic Order-Theoretic Characterization of the Weak Expectation Property

We prove the following characterization of the weak expectation property for operator systems in terms of an approximate version of Wittstock’s matricial Riesz separation property: an operator system S satisfies the weak expectation property if and only if

The classification problem for operator algebraic varieties and their multiplier algebras

M_{q}\left( S\right)

Strongly productive ultrafilters on semigroups

MqS satisfies the approximate matricial Riesz separation property for every

Polish groupoids and functorial complexity

q\in \mathbb {N}