This is a revision of the paper archived previously on August 22, 2002. It corrects a mistake in Sec. 8 concerning eccentricities of graphs. From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities are the vertice...

One of our results is a transfer principle between ultrapowers and reduced powers associated with the Fréchet ideal. Although motivated by Elliott classification programme, this result applies to any axiomatizable category. We also show that there exists nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$ such for every countable (or separable metric) structure $B$ in language quotient map f...

We continue in this article the abstract algebraic treatment of quantum sentential logics [39]. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultrapower, Frege relation, Leibniz congruence, Suszko congruence, Leibniz operator) are applied to quantum logic...

We show that Tsirelson’s problem concerning the set of quantum correlations and Connes’ embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg’s QWEP conjecture) are essentially equivalent. Specifically, Tsirelson’s problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the...

1. The hyperreals 3 1.1. Basic facts about the ordered real field 3 1.2. The nonstandard extension 4 1.3. Arithmetic in the hyperreals 5 1.4. The structure of N∗ 7 1.5. More practice with transfer 8 1.6. Problems 9 2. Logical formalisms for nonstandard extensions 10 2.1. Approach 1: The compactness theorem 11 2.2. Approach 2: The ultrapower construction 12 2.3. Problems 16 3. Sequences and seri...

It is shown that to every operator T in a general von Neumann factor M of type II1 and to every Borel set B in the complex plane C, one can associate a maximal, closed, T -invariant subspace, K = KT (B), affiliated with M, such that the Brown measure of T |K is concentrated on B. Moreover, K is T -hyperinvariant, and the Brown measure of PK⊥T |K⊥ is concentrated on C \ B. In particular, if T ∈ ...

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 the...

This note has two principal aims: to portray an essence of Non-Standard Analysis as a particular structure (which we call lim-rim), noting its interplay with the notion of ultrapower, and to present a construction of Non-Standard Analysis, viewed as a matter of mathematics, where the set A of non-standard elements of a set A – “the adjunction of all possible limits” is a “good” kind of lim-rim ...

We show that the class of 1-exact operator systems is not uniformly definable by a sequence of types. We use this fact to show that there is no finitary version of Arveson’s extension theorem. Next, we show that WEP is equivalent to a certain notion of existential closedness for C∗-algebras and use this equivalence to give a simpler proof of Kavruk’s result that WEP is equivalent to the complet...