arXiv subjects

Wee-Kee Tang

Publications and source records attributed to Wee-Kee Tang.

12 recordsLinked to original sources

arXiv2017

On ordinal ranks of Baire class functions

The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidnyánszky to Baire class $ξ$ functions for any countable ordinal $ξ\geq1$. In this paper, we answer two of the questions raised by them in their paper (Ranks on the Baire class $ξ$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $ξ\geq1,$ the ranks $β_ξ^{\ast}$ and $γ_ξ^{\ast}$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $β$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $β(fg)\leq ω^ξ$ whenever $β(g)\leqω^ξ$ for any countable ordinal $ξ.$

math.FA↗

arXiv2015

Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection between them that preserves order. We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to c_0, and for Banach lattices that contain a closed sublattice order isomorphic to c_0.

math.FA↗

arXiv2014

Nonlinear order isomorphisms on function spaces

Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between $A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order isomorphism between $A(X)$ and $A(Y)$ gives rise to an associated homeomorphism between $X$ and $Y$. This generalizes a classical result of Kaplansky concerning linear order isomorphisms between $C(X)$ and $C(Y)$ for compact Hausdorff $X$ and $Y$. The class of near vector lattices is introduced in order to extend the result further to noncompact spaces $X$ and $Y$. The main applications lie in the case when $X$ and $Y$ are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, "local" and "bounded local" versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.

math.FA↗

arXiv2009

Banach-Stone Theorems for maps preserving common zeros

Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a \emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E \to F$, $y \in Y$, and a function $h: Y \to X$. In this paper, we consider maps having the property: \cap^{k}_{i=1}Z(f_{i}) \neq\emptyset\iff\cap^{k}_{i=1}Z(Tf_{i}) \neq \emptyset, where $Z(f) = \{f = 0\}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including $C^{\infty}$), as Banach-Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that $X$ and $Y$ are realcompact spaces and $E$ and $F$ are Hausdorff topological vector lattices (respectively, $C^{*}$-algebras). Let $T: C(X,E) \to C(Y,F)$ be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. Then $X$ is homeomorphic to $Y$ and $E$ is lattice isomorphic (respectively, $C^{*}$-isomorphic) to $F$. Some results concerning the continuity of $T$ are also obtained.

math.FA↗

arXiv2007

Semilattice Structures of Spreading Models

Given a Banach space X, denote by SP_{w}(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SP_{w}(X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphic to SP_{w}(X) for some separable Banach space X.

math.FA↗

arXiv2006

More Mixed Tsirelson Spaces That Are Not Isomorphic To Their Modified Versions

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space T[(S_n,θ_{n})_{n=1}^{\infty}] and its modified version T_{M}[(S_{n},θ_{n})_{n=1}^{\infty}]. It is shown that these spaces are not isomorphic for a large class of parameters (θ_{n}).

math.FA↗

arXiv2005

Extension of Functions with Small Oscillation

A classical theorem of Kuratowski says that every Baire one function on a G_δsubspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that β_{Y}(f)<ω^α, α< ω_1, then f has an extension F onto X so that β_X(F)is not more than ω^α. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that β_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.

math.CA↗

arXiv2003

\ell^1-spreading models in subspaces of mixed Tsirelson spaces

We investigate the existence of higher order \ell^1-spreading models in subspaces of mixed Tsirelson spaces. For instance, we show that the following conditions are equivalent for the mixed Tsirelson space X=T[(θ_n,S_n)_{n=1}^{\infty}] (1)Every block subspace of $X$ contains an \ell^1-S_ω-spreading model, (2)The Bourgain \ell^1-index I_b(Y) = I(Y) > ω^ω for any block subspace Y of X, (3)\lim_m\limsup_nθ_{m+n}/θ_n > 0 and every block subspace Y of X contains a block sequence equivalent to a subsequence of the unit vector basis of X. Moreover, if one (and hence all) of these conditions holds, then X is arbitrarily distortable.

math.FA↗

arXiv2003

\ell ^1-spreading models in mixed Tsirelson space

Suppose that (F_n)_{n=1}^{\infty} is a sequence of regular families of finite subsets of N and (θ_n)_{n=1}^{\infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(θ_{n}, F_n)_{n=1}^{\infty}] is the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_0}, sup_n sup θ_n \sum_{i=1}^{j}||E_{i}x||}, where the last supremum is taken over all finite subsets E_{1},...,E_{j} of N such that E_1 < >... <E_j and {min E_1,...,min E_j} \in F_n. Necessary and sufficient conditions are obtained for the existence of higher order \ell ^1-spreading models in every subspace generated by a subsequence of the unit vector basis of T[(θ_{n}, F_n)_{n=1}^{\infty}.

math.FA↗

arXiv2001

The Bourgain ell ^1-index of mixed Tsirelson space

Suppose that (F_n)_{n=0}^{\infty} is a sequence of regular families of finite subsets of N such that F_0 contains all singletons, and (θ_n)_{n=1}^{\infty} is a nonincreasing null sequence in (0,1). In this paper, we compute the Bourgain \ell^1 - index of the mixed Tsirelson space T(F_0,(θ_n, F_n)_{n=1}^{\infty}). As a consequence, it is shown that if ηis a countable ordinal not of the form ω^ξfor some limit ordinal ξ, then there is a Banach space whose \ell^1-index is ω^η. This answers a question of Judd and Odell.

math.FA↗

arXiv2001

The $\ell ^{1}$-index of Tsirelson type spaces

If αand βare countable ordinals such that β\neq 0, denote by \tilde{T}_{α,β} the completion of $c_{00}$ with respect to the implicitly defined norm ||x|| = max{||x||_{c_{0}}, 1/2 sup \sum_{i=1}^{j}||E_{i}x||}, where the supremum is taken over all finite subsets E_{1},...,E_{j} of $\mathbb{N}$ such that $E_{1}<...<E_{j}$ and {min E_{1},...,min E_{j}} \in S_β. It is shown that the Bourgain $\ell^{1}$-index of \tilde{T}_{α,β} is ω^{α+β.ω}. In particular, if α=ω^{α_{1}}. m_{1}+...+ω^{α_{n}}. m_{n} in Cantor normal form and α_{n} is not a limit ordinal, then there exists a Banach space whose \ell^{1}-index is ω^α.

math.FA↗

arXiv2000

Functions of Baire class one

Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $β$ and the convergence index $γ$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$ for some countable ordinals $ξ_1$ and $ξ_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $\sup_nβ(f_n) \leq ω^{ξ_1}$ and $γ((f_n)) \leq ω^{ξ_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) \leq ω^{ξ_1}$ and $β(g) \leq ω^{ξ_2},$ then $β(fg) \leq ω^ξ,$ where $ξ=\max\{ξ_1+ξ_2, ξ_2+ξ_1}\}.$ These results do not assume the boundedness of the functions involved.

math.CA↗