The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidny{a}nszky to Baire class $xi$ functions for any countable ordinal $xigeq1$. In this paper, we answer two of the quest
ions raised by them in their paper (Ranks on the Baire class $xi$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $xigeq1,$ the ranks $beta_{xi}^{ast}$ and $gamma_{xi}^{ast}$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $beta$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $beta(fg)leq omega^{xi}$ whenever $beta(g)leqomega^{xi}$ for any countable ordinal $xi.$
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.
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
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 o
f 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}) eqemptysetiffcap^{k}_{i=1}Z(Tf_{i}) eq 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 Onal: 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) eqemptysetiff Z(Tf) eqemptyset. 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.
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 pa
ir 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.
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 inves
tigate the problem of comparing isomorphically the mixed Tsirelson space T[(S_n,theta_{n})_{n=1}^{infty}] and its modified version T_{M}[(S_{n},theta_{n})_{n=1}^{infty}]. It is shown that these spaces are not isomorphic for a large class of parameters (theta_{n}).
A classical theorem of Kuratowski says that every Baire one function on a G_delta subspace 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 beta(f). We prove a refinement of Kuratowskis theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that beta_{Y}(f)<omega^{alpha}, alpha < omega_1, then f has an extension F onto X so that beta_X(F)is not more than omega^{alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
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[(theta _n,S_n)_{n=1}^{infty}] (1)Every
block subspace of $X$ contains an ell^1-S_{omega}-spreading model, (2)The Bourgain ell^1-index I_b(Y) = I(Y) > omega^{omega} for any block subspace Y of X, (3)lim_mlimsup_ntheta_{m+n}/theta_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.
Suppose that (F_n)_{n=1}^{infty} is a sequence of regular families of finite subsets of N and (theta_n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T[(theta_{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 theta_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[(theta_{n}, F_n)_{n=1}^{infty}.
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 (theta _n)_{n=1}^{infty} is a nonincreasing null sequence in (0,1). In this paper, we compute the Bourgain ell^1 - in
dex of the mixed Tsirelson space T(F_0,(theta_n, F_n)_{n=1}^{infty}). As a consequence, it is shown that if eta is a countable ordinal not of the form omega^xi for some limit ordinal xi, then there is a Banach space whose ell^1-index is omega^eta . This answers a question of Judd and Odell.
