163 research outputs found

    Boundary integral representation of multipliers of fragmented affine functions and other intermediate function spaces

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    We develop a theory of abstract intermediate function spaces on a compact convex set XX 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 Af(X)A_f(X) of fragmented affine functions, to the space (Af(X))μ(A_f(X))^\mu, the monotone sequential closure of Af(X)A_f(X), 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 (Af(X))μ(A_f(X))^\mu 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 C(K)C(K)-spaces and the Josefson--Nissenzweig theorem

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    For a compact space KK, the Banach space C(K)C(K) is said to have the 1\ell_1-Grothendieck property if every weak* convergent sequence μn ⁣: nω\big\langle\mu_n\colon\ n\in\omega\big\rangle of functionals on C(K)C(K) such that μn1(K)\mu_n\in\ell_1(K) for every nωn\in\omega, is weakly convergent. Thus, the 1\ell_1-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that C(K)C(K) has the 1\ell_1-Grothendieck property if and only if there does not exist any sequence of functionals μn ⁣: nω\big\langle\mu_n\colon\ n\in\omega\big\rangle on C(K)C(K), with μn1(K)\mu_n\in\ell_1(K) for every nωn\in\omega, satisfying the conclusion of the classical Josefson--Nissenzweig theorem. We construct an example of a separable compact space KK such that C(K)C(K) has the 1\ell_1-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces KK their Banach spaces C(K)C(K) do not have the 1\ell_1-Grothendieck property.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0755

    Fra\"iss\'e Theory for Cuntz semigroups

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    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

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    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

    Point Degree Spectra of Represented Spaces

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    Generic properties of l_p-contractions and similar operator topologies

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    If XX is a separable reflexive Banach space, there are several natural Polish topologies on B(X)\mathcal{B}(X), the set of contraction operators on XX (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 B(X)\mathcal{B}(X). In this paper, we focus on p\ell_p\,-\,spaces, 1<p2<1<p\neq 2<\infty. We show that for some pairs of natural Polish topologies on B1(p)\mathcal B_1(\ell_p), the comeager sets are in fact the same; and our main result asserts that for p=3p=3 or 3/23/2 and in the real case, all topologies on B1(p)\mathcal B_1(\ell_p) 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 B1(p)\mathcal{B}_1 (\ell_p). 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

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    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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