163 research outputs found
Boundary integral representation of multipliers of fragmented affine functions and other intermediate function spaces
We develop a theory of abstract intermediate function spaces on a compact
convex set and study the behaviour of multipliers and centers of these
spaces. In particular, we provide some criteria for coincidence of the center
with the space of multipliers and a general theorem on boundary integral
representation of multipliers. We apply the general theory in several concrete
cases, among others to strongly affine Baire functions, to the space
of fragmented affine functions, to the space , the monotone
sequential closure of , to their natural subspaces formed by Borel
functions, or, in some special cases, to the space of all strongly affine
functions. In addition, we prove that the space is determined by
extreme points and provide a large number of illustrating examples and
counterexamples.Comment: 136 pages; we corrected one definition and expanded the introduction
a bi
Grothendieck -spaces and the Josefson--Nissenzweig theorem
For a compact space , the Banach space is said to have the
-Grothendieck property if every weak* convergent sequence
of functionals on such
that for every , is weakly convergent. Thus,
the -Grothendieck property is a weakening of the standard Grothendieck
property for Banach spaces of continuous functions. We observe that has
the -Grothendieck property if and only if there does not exist any
sequence of functionals on
, with for every , satisfying the
conclusion of the classical Josefson--Nissenzweig theorem. We construct an
example of a separable compact space such that has the
-Grothendieck property but it does not have the Grothendieck property.
We also show that for many classical consistent examples of Efimov spaces
their Banach spaces do not have the -Grothendieck property.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0755
Fra\"iss\'e Theory for Cuntz semigroups
We introduce a Fra\"iss\'e theory for abstract Cuntz semigroups akin to the
theory of Fra\"iss\'e categories developed by Kubi\'s. In particular, we show
that any (Cuntz) Fra\"iss\'e category has a unique Fra\"iss\'e limit which is
both universal and homogeneous. We also give several examples of such
categories and compute their Fra\"iss\'e limits. During our investigations, we
develop a general theory of Cauchy sequences and intertwinings in the category
Cu.Comment: 40 pages - minor change
Pushdown Normal-Form Bisimulation: A Nominal Context-Free Approach to Program Equivalence
We propose Pushdown Normal Form (PDNF) Bisimulation to verify contextual
equivalence in higher-order functional programming languages with local state.
Similar to previous work on Normal Form (NF) bisimulation, PDNF Bisimulation is
sound and complete with respect to contextual equivalence. However, unlike
traditional NF Bisimulation, PDNF Bisimulation is also decidable for a class of
program terms that reach bounded configurations but can potentially have
unbounded call stacks and input an unbounded number of unknown functions from
their context. Our approach relies on the principle that, in model-checking for
reachability, pushdown systems can be simulated by finite-state automata
designed to accept their initial/final stack content. We embody this in a
stackless Labelled Transition System (LTS), together with an on-the-fly
saturation procedure for call stacks, upon which bisimulation is defined. To
enhance the effectiveness of our bisimulation, we develop up-to techniques and
confirm their soundness for PDNF Bisimulation. We develop a prototype
implementation of our technique which is able to verify equivalence in examples
from practice and the literature that were out of reach for previous work
Generic properties of l_p-contractions and similar operator topologies
If is a separable reflexive Banach space, there are several natural
Polish topologies on , the set of contraction operators on
(none of which being clearly ``more natural'' than the others), and hence
several a priori different notions of genericity -- in the Baire category sense
-- for properties of contraction operators. So it makes sense to investigate to
which extent the generic properties, i.e. the comeager sets, really depend on
the chosen topology on . In this paper, we focus on
-spaces, . We show that for some pairs of
natural Polish topologies on , the comeager sets are in
fact the same; and our main result asserts that for or and in the
real case, all topologies on lying between the Weak
Operator Topology and the Strong Operator Topology share the same comeager
sets. Our study relies on the consideration of continuity points of the
identity map for two different topologies on . The
other essential ingredient in the proof of our main result is a careful
examination of norming vectors for finite-dimensional contractions of a special
type.Comment: 39
Descriptive Set Theory and Applications
The systematic study of Polish spaces within the scope of Descriptive Set Theory furnishes the working mathematician with powerful techniques and illuminating insights. In this thesis, we start with a concise recapitulation of some classical aspects of Descriptive Set Theory which is followed by a succint review of topological groups, measures and some of their associated algebras.The main application of these techniques contained in this thesis is the study of two families of closed subsets of a locally compact Polish groupG, namely U(G) - closed sets of uniqueness - and U0(G) - closed sets of extended uniqueness. We locate the descriptive set theoretic complexityof these families, proving in particular that U(G) is \Pi_1^1-complete whenever G/\overline{[G,G]} is non-discrete, thereby extending the existing literature regarding the abelian case. En route, we establish some preservation results concerning sets of (extended) uniqueness and their operator theoretic counterparts. These results constitute a pivotal part in the arguments used and entail alternative proofs regarding the computation of the complexity of U(G) and U0(G) in some classes of the abelian case.We study U(G) as a calibrated \Pi_1^1 \sigma-ideal of F(G) - for G amenable - and prove some criteria concerning necessary conditions for the inexistence of a Borel basis for U(G). These criteria allow us to retrieve information about G after examination of its subgroups or quotients. Furthermore, a sufficient condition for the inexistence of a Borel basis for U(G) is proven for the case when G is a product of compact (abelian or not) Polish groupssatisfying certain conditions.\ua0Finally, we study objects associated with the point spectrum of linear bounded operators T\in L(X) acting on a separable Banach space X. We provide a characterization of reflexivity for Banach spaces with an unconditional basis : indeed such space X is reflexive if and only if for all closed subspaces Y\subset X;Z\subset X^{\ast} and T\in 2 L(Y); S\in 2 L(Z) it holds that the point spectra \sigma_p(T); \sigma_p(S) are Borel. We study the complexity of sets prescribed by eigenvalues and prove a stability criterion for Jamison sequences
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