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.
In this paper we prove some results related to the problem of isomorphically classifying the complemented subspaces of $X_{p}$. We characterize the complemented subspaces of $X_{p}$ which are isomorphic to $X_{p}$ by showing that such a space must contain a canonical complemented subspace isomorphic to $X_{p}.$ We also give some characterizations of complemented subspaces of $X_{p}$ isomorphic to $\ell_{p}\oplus \ell_{2}.$
We provide a sufficient condition for the sum of a finite number of complemented subspaces of a Banach space to be complemented. Under this condition a formula for a projection onto the sum is given. We also show that the condition is sharp (in a certain sense). As applications, we get (1) sufficient conditions for the complementability of sums of marginal subspaces in $L^p$ and sums of tensor powers of subspaces in a tensor power of a Banach space and (2) quantitative results on stability of the complementability property of the sum of linearly independent subspaces.
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.
We show that a complemented subspace of a locally convex direct sum of an uncountable collection of Banach spaces is a locally convex direct sum of complemented subspaces of countable subsums. As a corollary we prove that a complemented subspace of a locally convex direct sum of arbitrary collection of $\ell_{1}(Γ)$-spaces is isomorphic to a locally convex direct sum of $\ell_{1}(Γ)$-spaces.
The complemented subspace problem asks, in general, which closed subspaces $M$ of a Banach space $X$ are complemented; i.e. there exists a closed subspace $N$ of $X$ such that $X=M\oplus N$? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some results on complemented subspaces, to give a history of the subject, and to present some open problems.
Understanding the complemented subspaces of $L_p$ has been an interesting topic of research in Banach space theory since 1960. 1999, Alspach proposed a systematic approach to classifying the subspaces of $L_p$ by introducing a norm given by partitions and weights. This paper shows that with Alspach Norm we are able to classify some complemented subspaces of $L_p, 2<p<\infty$.
It is shown that the Schreier space X admits a set of continuum cardinality whose elements are mutually incomparable complemented subspaces spanned by subsequences of the natural Schauder basis of X.
An isomorphic (\ell_1)-predual space (X) is constructed such that neither (X) is isomorphic to a subspace of (c_0), nor (C(ω^ω)) is isomorphic to a subspace of (X). It follows that (X) is not isomorphic to a complemented subspace of a (C(K)) space.
For every $n\geq 3,$ we construct an $n$-dimensional Banach space which is isometric to a subspace of $L_{1/2}$ but is not isometric to a subspace of $L_1.$ The isomorphic version of this problem (posed by S. Kwapien in 1969) is still open. Another example gives a Banach subspace of $L_{1/4}$ which does not embed isometrically in $L_{1/2}.$ Note that, from the isomorphic point of view, all the spaces $L_q$ with $q<1$ have the same Banach subspaces.
We prove that the subspace $c_0$ of sequences that converge to zero is not complemented in the space $ac_0$ of sequences that almost converge to zero. We proceed with applying the same approach to inclusion chain $c_0\subset A_0 \subset \ell_\infty$.
It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces. (This is a revised version of the paper by the same name formerly contained in the file ostrovskiicompsbsp.atx.)
We provide complete isomorphic invariance of a class of translation invariant complemented subspaces of L_p constructed by Bourgain, Rosenthal and Schechtman. We compute ordinal L_p-indices for this class. We further show that the isometric index of a tree subspace over a well founded tree is an invariance for the order of the tree. Finally we provide a dichotomy for the subspaces of L_p with small ordinal indices.
We first unify all notions of partial injectivity appearing in the literature ---(universal) separable injectivity, (universal) $\aleph$-injectivity --- in the notion of $(α, β)$-injectivity ($(α, β)_λ$-injectivity if the parameter $λ$ has to be specified). Then, extend the notion of space of universal disposition to space of universal $(α, β)$-disposition. Finally, we characterize the $1$-complemented subspaces of spaces of universal $(α, β)$-disposition as precisely the spaces $(α, β)_1$-injective.
Let $\mathcal{P}_{K} (^{n}E; F)$ (resp. $\mathcal{P}_{w} (^{n}E; F)$) the subspace of all $P\in \mathcal{P}(^{n}E; F)$ which are compact (resp. weakly continuous on bounded sets). We show that if $\mathcal{P}_{K} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{K} (^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$. Likewise we show that if $\mathcal{P}_{w} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{P}_{w}(^{n}E; F)$ is not complemented in $\mathcal{P}(^{n}E; F)$.
Many of the known complemented subspaces of L_p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well-known complemented subspaces of L_p. It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions and weights to be isomorphic to a subspace of L_p. Using this we define a space Y_n with norm…
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 …
We show that every Lorentz sequence space $d(\textbf{w},p)$ admits a 1-complemented subspace $Y$ distinct from $\ell_p$ and containing no isomorph of $d(\textbf{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\textbf{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\textbf{w}=(n^{-θ})_{n=1}^\infty$ ($0<θ<1$) as the $\ell_p$-sum of finite-dimensional copies of $d(\textbf{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal{L}(d(\textbf{w},p))$, of which only five …
If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A_0,A_1) such that A_0 and A_1 are isometric to $X\oplus V$, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.
Let X be an L_1-predual space and let K be a countable linearly independent subset of the extreme points of its closed dual ball. It is shown that if the norm-closed linear span Y of K is w^*-closed in X^*, then Y is the range of a w^*-continuous contractive projection in X^*. This result is applied in order to provide new and simpler proofs of the results of Lazar, Lindenstrauss and Zippin on the embedding of C(K) spaces into X.
Nonreflexive Banach spaces that are complemented in their bidual by an L-projection - like preduals of von Neumann algebras or the Hardy space $H^1$ - contain, roughly speaking, many copies of $l^1$ which are very close to isometric copies. Such $l^1$-copies are known to fail the fixed point property. Similar dual results hold for $c_0$.
Let X,Y be two Banach spaces ; in the first part of this work, we show that K(X,Y) contains a complemented copy of c0 if Y contains a copy of c0 and each bounded sequence in Y has a subsequece which is w* convergente. Afterward we obtain some results of M.Feder and G.Emmanuele: Finally in this part we study the relation between the existence of projection from L(X,Y) on K(X,Y) and the existence of pro- jection from K(X,Y ) on K(X,Y) if Y has the approximation property. In the second part we study the Radon-Nikodym property in L(X,Y):
Given the polynomial ideal $\mathcal{J}\circ\mathcal{P} (^{n}E; F)$, we prove that if $\mathcal{J}\circ\mathcal{P} (^{n}E; F)$ contains an isomorphic copy of $c_{0}$, then $\mathcal{J}\circ\mathcal{P} (^{n}E; F)$ is not complemented in $\mathcal{P} (^{n}E; F)$ for every closed operator ideal $\mathcal{J}\subset \mathcal{L}_{K}$ and every $n\in\mathbb{N}$. Likewise we show that if $\widehat{(\mathcal{J}\circ\mathcal{L})^{fac}}(^{n}E;F)$ contains an isomorphic copy of $c_{0}$, then $\widehat{(\mathcal{J}\circ\mathcal{L})^{fac}}(^{n}E;F)$ is not complemented in $\mathcal{P}(^{n}E; F)$ for every c…
We show the existence of a compact metric space $K$ such that whenever $K$ embeds isometrically into a Banach space $Y$, then any separable Banach space is linearly isometric to a subspace of $Y$. We also address the following related question: if a Banach space $Y$ contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space $X$, does it necessarily contain a subspace isometric to $X$? We answer positively this question when $X$ is a polyhedral finite-dimensional space, $c_0$ or $\ell_1$.
In this note we include two remarks about bounded ($\underline{not}$ necessarily contractive) linear projections on a von Neumann-algebra. We show that if $M$ is a von Neumann-subalgebra of $B(H)$ which is complemented in B(H) and isomorphic to $M \otimes M$ then $M$ is injective (or equivalently $M$ is contractively complemented). We do not know how to get rid of the second assumption on $M$. In the second part,we show that any complemented reflexive subspace of a $C^*$- algebra is necessarily linearly isomorphic to a Hilbert space.
Let $λ$ be an infinite cardinal number and let $\ell_\infty^c(λ)$ denote the subspace of $\ell_\infty(λ)$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite dimensional complemented subspaces of $\ell_\infty^c(λ)$, proving that they are isomorphic to $\ell_\infty^c(κ)$ for some cardinal number $κ$. Then we show that the Banach algebra of all bounded linear operators on $\ell_\infty^c(λ)$ or $\ell_\infty(λ)$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we …
In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of…
Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach space, $p>1,$ and $(X,σ)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is isometric to a subspace of $F.$ Moreover, every isometry $T$ from $E$ into $L_p(X;F)$ has the form $Te(x)=h(x)U(x)e, e\in E,$ where $h:X\rightarrow R$ is a measurable function and, for every $x\in X,$ $U(x)$ is an isometry from $E$ to $F.$
For $n\geq 2, p<2$ and $q>2,$ does there exist an $n$-dimensional Banach space different from Hilbert spaces which is isometric to subspaces of both $L_{p}$ and $L_{q}$? Generalizing the construction from the paper "Zonoids whose polars are zonoids" by R.Schneider we give examples of such spaces. Moreover, for any compact subset $Q$ of $(0,\infty)\setminus \{2k, k\in N\},$ we can construct a space isometric to subspaces of $L_{q}$ for all $q\in Q$ simultaneously. This paper requires vanilla.sty
In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an extension operator. The space C(L) can be replaced by its subspace C(L|F) consisting of functions that vanish on a closed subset F of L. We also study the class of spaces having the extension property, establishing some closure results for this class and relating it to other classes of compact spaces.
We construct some separable infinite dimensional homogeneous Hilbertian operator spaces which generalize the row and column spaces R and C. We show that separable infinite-dimensional Hilbertian JC*-triples are completely isometric to an element of the set of (infinite) intersections of these spaces. This set includes the operator spaces R, C, their intersection, and the space spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that these new spaces are completely isometric to the space of creation (resp. annihilation) operators on the anti-symmetric tensors o…
We consider the Schatten spaces S^p in the framework of operator space theory and for any $1\leq p\not=2<\infty$, we characterize the completely 1-complemented subspaces of S^p. They turn out to be the direct sums of spaces of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a description of all 1-complemented subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L^p-spaces. Also we show that for any $n\geq 2$…
We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,π)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.
The spaces $W_α$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $α\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_α$, where the norm of $α$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have $$ r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \l…
We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $\ell_p(X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space $\ell…
The Rademacher functions are investigated in the Morrey spaces M(p,w) on [0,1] for 1 \le p <\infty and weight w being a quasi-concave function. They span l_2 space in M(p,w) if and only if the weight w is smaller than the function log_2^{-1/2}(2/t) on (0,1). Moreover, if 1 < p < \infty the Rademacher sunspace R_p is complemented in M(p,w) if and only if it is isomorphic to l_2. However, the Rademacher subspace is not complemented in M(1,w) for any quasi-concave weight w. In the last part of the paper geometric structure of Rademacher subspaces in Morrey spaces M(p,w) is described. It turns out…
In this paper we provide a study of quaternionic inner product spaces. This includes ortho-complemented subspaces, fundamental decompositions as well as a number of results of topological nature. Our main purpose is to show that a uniformly positive subspace in a quaternionic Krein space is ortho-complemented, and this leads to our choice of the results presented in the paper.
Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of $\ell_p^n$, $p$ an even integer, needed to almost isometrically contain all $k$-dimensional subspaces of $L_p$.
The operator spaces $H_n^k$ $1\le k\le n$, generalizing the row and column Hilbert spaces, and arising in the authors' previous study of contractively complemented subspaces of $C^*$-algebras, are shown to be homogeneous and completely isometric to a space of creation operators on a subspace of the anti-symmetric Fock space. The completely bounded Banach-Mazur distance from $H_n^k$ to row or column space is explicitly calculated.
We prove a number of results concerning the embedding of a Banach lattice $X$ into an r.i. space $Y$. For example we show that if $Y$ is an r.i. space on $[0,\infty)$ which is $p$-convex for some $p>2$ and has nontrivial concavity then any Banach lattice $X$ which is $r$-convex for some $r>2$ and embeds into $Y$ must embed as a sublattice. Similar conclusions can be drawn under a variety of hypotheses on $Y$; if $X$ is an r.i. space on $[0,1]$ one can replace the hypotheses of $r$-convexity for some $r>2$ by $X\neq L_2.$ We also show that if $Y$ is an order-continuous Banach lattice which cont…
Dronov and Kaplitzki showed that every complemented subspace of a nuclear Köthe space E with a regular basis of type ($d_1$) has a basis so, in particular, solving the long standing problem whether any complemented subspace of the space (s) of rapidly decreasing sequences has a basis. We present a slightly modified version of their proof which shows that the range of every closed-range operator in $E$ has a basis
Let $(E,F)$ be a pair of Fréchet spaces. In this paper, we discuss whether a certain property $P$ enjoyed by both $E$ and $F$ is also satisfied by the complete tensor product $E \widehat{\otimes}_π F$. Specifically we focus on the two properties generalizing Köthe spaces: the normability condition called the property $(y)$, and the property of smallness of bounded subsets of Fréchet spaces up to a complemented Banach subspace in connection with the problem of topologies of A. Grothendieck. We also consider the stability of some well known equivalences of operator classes with an application in…
