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Paul Bankston - One of the best experts on this subject based on the ideXlab platform.
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MAPPING PROPERTIES OF CO-EXISTENTIALLY CLOSED CONTINUA
2016Co-Authors: Paul Bankston, Communicated Charles HagopianAbstract:Abstract. A Continuous Surjection between compacta is called co-existential if it is the second of two maps whose composition is a standard ultracopower projection. A continuum is called co-existentially closed if it is only a co-existential image of other continua. This notion is not only an exact dual of Abraham Robinson’s existentially closed structures in model theory, it also parallels the definition of other classes of continua defined by what kinds of Continuous images they can be. In this paper we continue our study of co-existentially closed continua, especially how they (and related continua) behave in certain mapping situations. 1. introduction By a compactum we mean a compact Hausdorff space, a continuum is a connected compactum. A subcompactum (resp., subcontinuum) of a space is just a subspace that is itself a compactum (resp., continuum). Given a compactum X and an ultrafilter D on an index set I (i.e., D is a maximal filter in the Boolean power set algebra of I), the ultracopower of X via D is denoted XI \ D. One easy way to describe this construction is to regard I as a discrete space, letting p: X × I → X and q: X × I → I be the standard projection maps. Applying the Stone-Čech compactification functor β ( ) (see, e.g., [23, 24]), we regard D as a point in β(I) and define the ultracopower to be the inverse image of D under qβ. We denote by pX,D the restriction of p
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Not every co-existential map is confluent
2015Co-Authors: Paul Bankston, Communicated Charles HagopianAbstract:Abstract. A Continuous Surjection between compacta is co-existential if it is the second of two maps whose composition is a standard ultracopower projection. Co-existential maps are always weakly confluent, and are even monotone when the range space is locally connected; so it is a natural ques-tion to ask whether they are always confluent. Here we give a negative answer. This is an interesting question, mainly because of the fact that most the-orems about confluent maps have parallel versions for co-existential maps— notably, both kinds of maps preserve hereditary indecomposability. Where the known parallels break down is in the question of chainability. It is a cel-ebrated open problem whether confluent maps preserve chainability, or even being a pseudo-arc; however, as has recently been shown [7], co-existential maps do indeed preserve both these properties. 1. introduction Co-existential maps are defined using topological ultracopowers in an exact mir-roring of how one characterizes the existential embeddings of model theory in terms of ultrapowers. (See, e.g., [3] for a full explanation.) Briefly, if X is a com-pactum (i.e., a compact Hausdorff space) and D is an ultrafilter on a set I (viewed as a discrete topological space), then we let p: X × I → X and q: X × I → I be the standard projection maps. The D-ultracopower of X is denoted XI\D, and is the inverse image of the point D ∈ β(I) with respect to the Stone-Čech lif
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MAPPING PROPERTIES OF CO-EXISTENTIALLY CLOSED CONTINUA
2015Co-Authors: Paul BankstonAbstract:Abstract. A Continuous Surjection between compacta is called co-existential if it is the second of two maps whose composition is a standard ultracopower projection. A continuum is called co-existentially closed if it is only a co-existential image of other continua. This notion is not only an exact dual of Abraham Robinson’s existentially closed structures in model theory, it also parallels the definition of other classes of continua defined by what kinds of Continuous images they can be. In this paper we continue our study of co-existentially closed continua, especially how they (and related continua) behave in certain mapping situations. 1. introduction By a compactum we mean a compact Hausdorff space, a continuum is a con-nected compactum. A subcompactum (resp., subcontinuum) of a space is just a subspace that is itself a compactum (resp., continuum). Given a compactumX and an ultrafilter D on an index set I (i.e., D is a maximal filter in the Boolean power set algebra of I), the ultracopower of X via D i
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CO-ELEMENTARY EQUIVALENCE, CO-ELEMENTARY MAPS, AND GENERALIZED ARCS
2008Co-Authors: Paul Bankston, Communicated Andreas R. BlassAbstract:Abstract. By a generalized arc we mean a continuum with exactly two non-separating points; an arc is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if f: X → Y is a function between compacta and if X is an arc, then f is a co-elementary map if and only if Y is an arc and f is a monotone Continuous Surjection
Bankston Paul - One of the best experts on this subject based on the ideXlab platform.
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Not Every Co-existential Map is Confluent
e-Publications@Marquette, 2010Co-Authors: Bankston PaulAbstract:A Continuous Surjection between compacta is co-existential if it is the second of two maps whose composition is a standard ultracopower projection. Co-existential maps are always weakly confluent, and are even monotone when the range space is locally connected; so it is a natural question to ask whether they are always confluent. Here we give a negative answer. This is an interesting question, mainly because of the fact that most theorems about confluent maps have parallel versions for co-existential maps---notably, both kinds of maps preserve hereditary indecomposability. Where the known parallels break down is in the question of chainability. It is a celebrated open problem whether confluent maps preserve chainability, or even being a pseudo-arc; however, as has recently been shown, co-existential maps do indeed preserve both these properties
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Mapping Properties of Co-existentially Closed Continua
e-Publications@Marquette, 2005Co-Authors: Bankston PaulAbstract:A Continuous Surjection between compacta is called co-existential if it is the second of two maps whose composition is a standard ultracopower projection. A continuum is called co-existentially closed if it is only a co-existential image of other continua. This notion is not only an exact dual of Abraham Robinson\u27s existentially closed structures in model theory, it also parallels the definition of other classes of continua defined by what kinds of Continuous images they can be. In this paper we continue our study of co-existentially closed continua, especially how they (and related continua) behave in certain mapping situations
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Co-elementary equivalence, co-elementary maps, and generalized arcs
1994Co-Authors: Bankston PaulAbstract:By a {\bf generalized arc\/} we mean a continuum with exactly two non-separating points; an {\bf arc} is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if $f:X \to Y$ is a function between compact Hausdorff spaces and if $X$ is an arc, then $f$ is a co-elementary map if and only if $Y$ is an arc and $f$ is a monotone Continuous Surjection
Spurný Jiří - One of the best experts on this subject based on the ideXlab platform.
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Preserving affine Baire classes by perfect affine maps
'Informa UK Limited', 2016Co-Authors: Kalenda, Ondřej F.k., Spurný JiříAbstract:Let φ: X → Y be an affine Continuous Surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine Continuous functions on Y in the space of real-valued affine Continuous functions on X is complemented. We show that if F is a topological vector space, then f : Y → F is of affine Baire class α whenever the composition f ○ φ is of affine Baire class α. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.Keywords: Vector-valued Baire function, strongly affine functio
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Preserving affine Baire classes by perfect affine maps
'National Inquiry Services Center (NISC)', 2015Co-Authors: Kalenda, Ondřej F. K., Spurný JiříAbstract:Let $\varphi\colon X\to Y$ be an affine Continuous Surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine Continuous functions on $Y$ in the space of real-valued affine Continuous functions on $X$ is complemented. We show that if $F$ is a topological vector space, then $f\colon Y\to F$ is of affine Baire class $\alpha$ whenever the composition $f\circ\varphi$ is of affine Baire class $\alpha$. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.Comment: 10 page
D. R. Pitts - One of the best experts on this subject based on the ideXlab platform.
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ERRATUM Erratum to: The algebraic structure of non-commutative analytic Toeplitz algebras
2015Co-Authors: K. R. Davidson, D. R. PittsAbstract:The algebraic structure of the non-commutative analytic Toeplitz algebra Ln is devel-oped in the original article. Some of the results fail for the case n = ∞, and this implies that certain other results are not established in this case. In Theorem 3.2 of the original article, we showed there is Continuous Surjection πn,k from Repk(Ln), the space of completely contractive representations of Ln into the k × k matrices Mk, onto the closed unit ball Bn,k of Rn(Mk) by evaluation at the generators. It is further claimed that if T = [T1,..., Tn] ∈ Rn(Mk) with ‖T ‖ < 1, then there is a unique representation in π−1n,k (T). Further information is obtained for k = 1 in Theorem 3.3 of the original article. Our proof of these results is valid for n < ∞, however, for n = ∞ the uniqueness claim is incorrect. An example due to Michael Hartz (see [2, Example 2.4]) shows that π−1∞,1(0) is very large—it contains a copy of the βN\N. The difficulty in the proof of Theorems 3.2 and 3.3 of the original article stems from the use of the factorization A = W X used in Lemma 3.1 of the original article. In the case n = ∞, this factorization comes from Corollary 2.9. The problem is that the infinite sum in Corollary 2.9 converges in the strong topology, not the norm topology, so that when the representation is not strongly Continuous (or equivalently, The online version of the original article can be found under doi:10.1007/s002080050188
Mineyama Ryosuke - One of the best experts on this subject based on the ideXlab platform.
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Cannon-Thurston maps for Coxeter groups including affine special subgroups
2014Co-Authors: Mineyama RyosukeAbstract:For a Coxeter group $W$ we have an associating bi-linear form $B$ on a real vector space. We assume that $B$ has the signature $(n-1,1)$. In this case we have the Cannon-Thurston map for $W$, that is, a $W$-equivariant Continuous Surjection from the Gromov boundary of $W$ to the limit set of $W$. We focus on the case where Coxeter groups contain affine special subgroups.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1312.317
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Cannon-Thurston maps for Coxeter groups with signature $(n-1,1)$
2014Co-Authors: Mineyama RyosukeAbstract:For a Coxeter group $W$ we have an associating bi-linear form $B$ on suitable real vector space. We assume that $B$ has the signature $(n-1,1)$ and all the bi-linear form associating rank $n' (\ge 3)$ Coxeter subgroups generated by subsets of $S$ has the signature $(n',0)$ or $(n'-1,1)$. Under these assumptions, we see that there exists the Cannon-Thurston map for $W$, that is, the $W$-equivariant Continuous Surjection from the Gromov boundary of $W$ to the limit set of $W$. To see this we construct an isometric action of $W$ on an ellipsoid with the Hilbert metric. As a consequence, we see that the limit set of $W$ coincides with the set of accumulation points of roots of $W$.Comment: 24 page