111 research outputs found
Some Representation Theorem for nonreflexive Banach space ultrapowers under the Continuum Hypothesis
In this paper it will be shown that assuming the Continuum Hypothesis (CH)
every nonreflexive Banach space ultrapower is isometrically isomorphic to the
space of continuous, bounded and real-valued functions on the Parovicenko
space. This Representation Theorem will be helpful in proving some facts from
geometry and topology of nonreflexive Banach space ultrapowers.Comment: 12 page
Possible Size of an ultrapower of omega
Let omega be the first infinite ordinal (or the set of all natural numbers)
with the usual order <. In section 1 we show that, assuming the consistency of
a supercompact cardinal, there may exist an ultrapower of omega, whose
cardinality is (1) a singular strong limit cardinal, (2) a strongly
inaccessible cardinal. This answers two questions in [CK], modulo the
assumption of supercompactness. In section 2 we construct several
lambda-Archimedean ultrapowers of omega under some large cardinal assumptions.
For example, we show that, assuming the consistency of a measurable cardinal,
there may exist a lambda-Archimedean ultrapower of omega for some uncountable
cardinal lambda. This answers a question in [KS], modulo the assumption of
measurability
A Survey of Ultraproduct Constructions in General Topology
We survey various attempts to transport the ultraproduct construction from
the realm of model theory to that of general topology
Easton's Theorem for Ramsey and Strongly Ramsey cardinals
We show that, assuming GCH, if is a Ramsey or a strongly Ramsey
cardinal and is a class function on the regular cardinals having a closure
point at and obeying the constraints of Easton's theorem, namely,
for and \alpha<\cf(F(\alpha)),
then there is a cofinality preserving forcing extension in which
remains Ramsey or strongly Ramsey respectively and for
every regular cardinal .Comment: 21 page
Natural numerosities of sets of tuples
We consider a notion of "numerosity" for sets of tuples of natural numbers,
that satisfies the five common notions of Euclid's Elements, so it can agree
with cardinality only for finite sets. By suitably axiomatizing such a notion,
we show that, contrasting to cardinal arithmetic, the natural "Cantorian"
definitions of order relation and arithmetical operations provide a very good
algebraic structure. In fact, numerosities can be taken as the non-negative
part of a discretely ordered ring, namely the quotient of a formal power series
ring modulo a suitable ("gauge") ideal. In particular, special numerosities,
called "natural", can be identified with the semiring of hypernatural numbers
of appropriate ultrapowers of N.Comment: 17 page
Boolean ultrapowers, the Bukovsky-Dehornoy phenomenon, and iterated ultrapowers
We show that while the length iterated ultrapower by a normal
ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing,
it is consistent that no iteration of length greater than (of the same
ultrafilter and its images) is a Boolean ultrapower. For longer iterations,
where different ultrafilters are used, this is possible, though, and we give
Magidor forcing and a generalization of Prikry forcing as examples. We refer to
the discovery that the intersection of the finite iterates of the universe by a
normal measure is the same as the generic extension of the direct limit model
by the critical sequence as the Bukovsky-Dehornoy phenomenon, and we develop a
sufficient criterion (the existence of a simple skeleton) for when a version of
this phenomenon holds in the context of Boolean ultrapowers. Assuming that the
canonical generic filter over the Boolean ultrapower model has what we call a
continuous representation, we show that the Boolean model consists precisely of
those members of the intersection model that have continuously and eventually
uniformly represented codes
Set theory and C*-algebras
We survey the use of extra-set-theoretic hypotheses, mainly the continuum
hypothesis, in the C*-algebra literature. The Calkin algebra emerges as a basic
object of interest.Comment: 16 pages; to appear in Bull. Symb. Logi
Set theoretical forcing in quantum mechanics and AdS/CFT correspondence
We show unexpected connection of Set Theoretical Forcing with Quantum
Mechanical lattice of projections over some separable Hilbert space. The basic
ingredient of the construction is the rule of indistinguishability of Standard
and some Nonstandard models of Peano Arithmetic. The ingeneric reals introduced
by M. Ozawa will correspond to simultaneous measurement of incompatible
observables. We also discuss some results concerning model theoretical analysis
of Small Exotic Smooth Structures on topological 4-space. Forcing appears
rather naturally in this context and the rule of indistinguishability is
crucial again. As an unexpected application we are able to approach Maldacena
Conjecture on correspondence in the case of AdS_5xS^5 and Super YM
Conformal Field Theory in 4 dimensions. We conjecture that there is possibility
of breaking Supersymetry via sources of gravity generated in 4 dimensions by
exotic smooth structures on R^4 emerging in this context.Comment: 16 pages, 1 eps figure, LaTeX 2e. Presented at QCS02 held in Ustron,
Poland on September 2002, to appear in Int. J. of Theor. Phy
On separably injective Banach spaces and Corrigendum to "On separably injective Banach spaces" [Adv. Math. 234 (2013) 192--216]
In this paper we deal with two weaker forms of injectivity which turn out to
have a rich structure behind: separable injectivity and universal separable
injectivity. We show several structural and stability properties of these
classes of Banach spaces. We provide natural examples of (universally)
separably injective spaces, including ultraproducts built
over countably incomplete ultrafilters, in spite of the fact that these
ultraproducts are never injective. We obtain two fundamental characterizations
of universally separably injective spaces: a) A Banach space is universally
separably injective if and only if every separable subspace is contained in a
copy of inside . b) A Banach space is universally
separably injective if and only if for every separable space one has
\Ext(\ell_\infty/S, E)=0. The final Section of the paper focuses on special
properties of 1-separably injective spaces. Lindenstrauss\ obtained in the
middle sixties a result that can be understood as a proof that, under the
continuum hypothesis, 1-separably injective spaces are 1-universally separably
injective; he left open the question in {\sf ZFC}. We construct a consistent
example of a Banach space of type which is 1-separably injective but not
1-universally separably injective. We show that, under the continuum
hypothesis, "to be universally separably injective" is not a -space
property, as we wrongly claimed in the paper mentioned in the title.Comment: The first paper was published in [Adv. Math. 234 (2013) 192--216].
The corrigendum is to appear in Adv. in Mat
Extremal structure in ultrapowers of Banach spaces
Given a bounded convex subset C of a Banach space X and a free ultrafilter U, we study which points (xi)U are extreme points of the ultrapower CU in XU. In general, we obtain that when { xi} is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then (xi)U is an extreme point (respectively denting point, strongly exposed point) of CU. We also show that every extreme point of CU is strongly extreme, and that every point exposed by a functional in (X*)U is strongly exposed, provided that U is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of CU in the case that C is a super weakly compact or uniformly convex set. © 2022, The Author(s)
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