71 research outputs found
Constructive topology of bishop spaces
The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS.
Instead of having a common space-structure on a set X and R, where R denotes the set of
constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F),
where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R.
The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions,
the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a
constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces.
The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient,
the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact,
the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces.
The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively
the Urysohn extension theorem for Bishop spaces.
The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included
Order theory and interpolation in operator algebras
We continue our study of operator algebras with and contractive approximate
identities (cais). In earlier papers we have introduced and studied a new
notion of positivity in operator algebras, with an eye to extending certain
C*-algebraic results and theories to more general algebras. Here we continue to
develop this positivity and its associated ordering, proving many foundational
facts. We also give many applications, for example to noncommutative topology,
noncommutative peak sets, lifting problems, peak interpolation, approximate
identities, and to order relations between an operator algebra and the
C*-algebra it generates. In much of this it is not necessary that the algebra
have an approximate identity. Many of our results apply immediately to function
algebras, but we will not take the time to point these out, although most of
these applications seem new.Comment: 27 pages. arXiv admin note: substantial text overlap with
arXiv:1308.272
Noncommutative topology and Jordan operator algebras
Jordan operator algebras are norm-closed spaces of operators on a Hilbert
space with for all . We study noncommutative topology,
noncommutative peak sets and peak interpolation, and hereditary subalgebras of
Jordan operator algebras. We show that Jordan operator algebras present perhaps
the most general setting for a `full' noncommutative topology in the
C*-algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for
not necessarily selfadjoint algebras by the authors with Read, Hay and other
coauthors. Our breakthrough relies in part on establishing several strong
variants of C*-algebraic results of Brown relating to hereditary subalgebras,
proximinality, deeper facts about for a left ideal in a C*-algebra,
noncommutative Urysohn lemmas, etc. We also prove several other approximation
results in -algebras and various subspaces of -algebras, related to
open and closed projections, and technical -algebraic results of Brown.Comment: Revision, many typos corrected and exposition improved in places.
Section 2 expanded with some applications of the main theorem of that sectio
A Bishop-Phelps-Bollobas Type Theorem for uniform algegras
This paper is devoted to showing that Asplund operators with range in a uniform Banach algebra have
the Bishop¿Phelps¿Bollobas property, i.e., they are approximated by norm attaining Asplund operators
at the same time that a point where the approximated operator almost attains its norm is approximated
by a point at which the approximating operator attains it. To prove this result we use the weak*-to-norm
fragmentability of weak*-compact subsets of the dual of Asplund spaces and we need to observe a Urysohn type result producing peak complex-valued functions in uniform algebras that are small outside a given open set and whose image is inside a Stolz region.This research was partially supported by MEC and FEDER projects MTM2008-05396 and MTM2011-25377. The research of the second author was also partially supported by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project PAID-06-09-2829).Cascales, B.; Guirao Sánchez, AJ.; Kadets, V. (2013). A Bishop-Phelps-Bollobas Type Theorem for uniform algegras. Advances in Mathematics. 240:370-382. https://doi.org/10.1016/j.aim.2013.03.005S37038224
On the class of continuous images of non-commutative Valdivia compacta
We investigate the class of continuous images of non-commutative Valdivia
compact spaces, in particular its subclass of weakly non-commutative Cor- son
countably compact spaces. A key tool is the study of non-commutative Corson
countably compact spaces and their stability. The results are the non-
commutative version of results by O. Kalenda (2003). Moreover, we present a
study of retractional skeletons on Aleksandrov duplicates of ordinal spaces
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