A disjointness type property of conditional expectation operators

We give a characterization of conditional expectation operators through a disjointness type property similar to band preserving operators. We say that the operator $T:X\to X$ on a Banach lattice $X$ is semi band preserving if and only if for all $f, g \in X$, $f \perp Tg$ implies that $Tf \perp Tg$. We prove that when $X$ is a purely atomic Banach lattice, then an operator $T$ on $X$ is a weighted conditional expectation operator if and only if $T$ is semi band preserving.Comment: 11 page

1-complemented subspaces of spaces with 1-unconditional bases

We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space $X$ is not isomorphic to $\ell_p$ for some $1 \leq p < \infty$ then the only subspaces of $X$ which are $1$-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients

On the structure of level sets of uniform and Lipschitz quotient mappings from Rn{\mathbb{R}}^n to R{\mathbb{R}}

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient maps from $R^n\to R$. We show that if $f:R^n\to R$, $n\geq 2$, is a uniform quotient map then for every $t\in R$, $f^{-1}(t)$ has a bounded number of components, each component of $f^{-1}(t)$ separates $R^n$ and the upper bound of the number of components depends only on $n$ and the moduli of co-uniform and uniform continuity of $f$. Next we obtain a characterization of the form of any closed, hereditarily locally connected, locally compact, connected set with no end points and containing no simple closed curve, and we apply it to describe the structure of level sets of co-Lipschitz uniformly continuous mappings $f:R^2\to R$. We prove that all level sets of any co-Lipschitz uniformly continuous map from $R^2$ to $R$ are locally connected, and we show that for every pair of a constant $c>0$ and a function $\Omega$ with $\lim_{r\to 0}\Omega(r)=0$, there exists a natural number $M=M(c,\Omega)$, so that for every co-Lipschitz uniformly continuous map $f:R^2\to R$ with a co-Lipschitz constant $c$ and a modulus of uniform continuity $\Omega$, there exists a natural number $n(f)\le M$ and a finite set $T_f\subset R$ with \card(T_f)\leq n(f)-1 so that for all $t\in R\setminus T_f$, $f^{-1}(t)$ has exactly $n(f)$ components, $R^2\setminus f^{-1}(t)$ has exactly $n(f)+1$ components and each component of $f^{-1}(t)$ is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of $f^{-1}(s)$ for $s\in T_f$ are also described - they have a finite graph structure. We give an example of a uniform quotient map from $R^2\to R$ which has non-locally connected level sets.Comment: 34 pages, 10 figure

On isometric stability of complemented subspaces of Lp{L_p}

We show that Rudin-Plotkin isometry extension theorem in $L_p$ implies that when $X$ and $Y$ are isometric subspaces of $L_p$ and $p$ is not an even integer, $1 \leq p < \infty$, then $X$ is complemented in $L_p$ if and only if $Y$ is; moreover the constants of complementation of $X$ and $Y$ are equal. We provide examples demonstrating that this fact fails when $p$ is an even integer larger than 2

A note on Banach--Mazur problem

We prove that if $X$ is a real Banach space, with $\dim X\geq 3$, which contains a subspace of codimension 1 which is 1-complemented in $X$ and whose group of isometries is almost transitive then $X$ is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.Comment: 8 pages, 2 figures but one of the figures doesn't run well in TeX so it is not included here. The ps file of this paper which includes all figures is available at http://www.users.muohio.edu/randrib/bm3.ps. to appear in Glasgow J. Math. (2002

Isometric classification of norms in rearrangement-invariant function spaces

Suppose that a real nonatomic function space on $[0,1]$ is equipped with two re\-arran\-ge\-ment-invariant norms $\|\cdot\|$ and $|||\cdot|||$. We study the question whether or not the fact that $(X,\|\cdot\|)$ is isometric to $(X,|||\cdot|||)$ implies that $\|f\|= |||f|||$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, resp. Lorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class of Orlicz spaces where it fails. We provide a complete description of Orlicz functions $\varphi \neq\psi$ with the property that $L_\varphi$ and $L_\psi$ are isometric

Contractive projections in Orlicz sequence spaces

We characterize norm one complemented subspaces of Orlicz sequence spaces $\ell_M$ equipped with either Luxemburg or Orlicz norm, provided that the Orlicz function $M$ is sufficiently smooth and sufficiently different from the square function. This paper concentrates on the more difficult real case, the complex case follows from previously known results.Comment: 14 page

Injective isometries in Orlicz spaces

We show that injective isometries in Orlicz space $L_M$ have to preserve disjointness, provided that Orlicz function $M$ satisfies $\Delta_2$-condition, has a continuous second derivative $M''$, satisfies another ``smoothness type'' condition and either $\lim_{t\to0} M''(t) = \infty$ or $M''(0) = 0$ and $M''(t)>0$ for all $t>0$. The fact that surjective isometries of any rearrangement-invariant function space have to preserve disjointness has been determined before. However dropping the assumption of surjectivity invalidates the general method. In this paper we use a differential technique.Comment: 20 pages, 2 figures, to appear in the Proceedings of the Third Conference on Function Spaces held in Edwardsville in May 1998, Contemporary Mat

One-complemented subspaces of real sequence spaces

Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm

Contractive projections and isometries in sequence spaces

We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries