Let $\M$ be a semi-finite von Neumann algebra equipped with a faithful normal trace $τ$. We study the subspace structures of non-commutative Lorentz spaces $L_{p,q}(\M, τ)$, extending results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, $\ell_p$ can not be embedded into $L_{p,q}(\M, τ)$. As applications, we prove that for $0<p<\infty$ with $p \neq 2$ then $\ell_p$ cannot be strongly embedded into $L_p(\M,τ)$. Thus providing a non-commutative extension of a result of Kalton for $0<p<1$ and a result of Rosenthal for $1\leq p <2$ on $L_p[0,1]$.
In this paper we investigate the conditional expectation on the non-commutative $H^{(r,s)}_{p}(\mathcal A;\ell_{\infty})$ and $H_{p}(\mathcal A;\ell_{1})$ spaces associated with semifinite subdiagonal algebra, and prove the contractibility of the underlying conditional expectation on these spaces.
We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.
We study some structural aspects of the subspaces of the non-commutative (Haagerup) L_p-spaces associated with a general (non necessarily semi-finite) von Neumann algebra A. If a subspace X of L_p(A) contains uniformly the spaces \ell_p^n, n>= 1, it contains an almost isometric, almost 1-complemented copy of \ell_p. If X contains uniformly the finite dimensional Schatten classes S_p^n, it contains their \ell_p-direct sum too. We obtain a version of the classical Kadec-Pel czynski dichotomy theorem for L_p-spaces, p>= 2. We also give operator space versions of these results. The proofs are base…
Let $A$ be a real commutative Banach algebra with unity. Let $a_0\in A\setminus\{0\}$. Let $\mathbb Z a_0:=\{na_0\}_{n\in \mathbb Z}$. Then, $\mathbb Z a_0$ is a discrete subgroup of $A$. For any $n\in \mathbb Z$, the Frechet derivative of the mapping $$x \, \in \, A \ \ \ \mapsto \ \ \ x+na_0 \, \in \, A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(\mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x \, \in \, A \ \ \ \mapsto \ \ \ x+\mathbb Z a_0 \, \in \, A/(\mathbb Z a_0)$$ is an $A$-map. We call …
The isometric embeddings $\ell_{2;K}^m\to\ell_{p;K}^n$ ($m\geq 2$, $p\in 2\N$) over a field $K\in{R, C, H}$ are considered, and an upper bound for the minimal $n$ is proved. In the commutative case ($K\neq H$) the bound was obtained by Delbaen, Jarchow and Pełczy{ń}ski (1998) in a different way.
Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach embed in L^p(M) for N infinite, M finite, 1 < or = p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class C_p embeds in L^p(N) for N infinite). Theorem: Let 1 < or = p < 2 and let X be a Banach space with a spanning set (x_{ij}) so that for some C < or = 1: (i) any row or column is C-equivalent to the usual ell^2-basis; (ii) (x_{i_k,j_k}) is C-equivalent to the usual ell^p-basis, for any i_1 < i_2 < ... and j_1 < j_2 < ... . Then X is not isomorphic to a subspace…
James Tree Space ($\mathcal{JT}$), introduced by R. James, is the first Banach space constructed having non-separable conjugate and not containing $\ell^1$. James actually proved that every infinite dimensional subspace of $\mathcal{JT}$ contains a Hilbert space, which implies the $\ell^1$ non-embedding. In this expository article, we present a direct proof of the $\ell^1$ non-embedding, using Rosenthal's $\ell^1$- Theorem and some measure theoretic arguments, namely Riesz's Representation Theorem.
A Banach space is said to have the ball-covering property (abbreviated BCP) if its unit sphere can be covered by countably many closed, or equivalently, open balls off the origin. Let $K$ be a locally compact Hausdorff space and $X$ be a Banach space. In this paper, we give a topological characterization of BCP, that is, the continuous function space $C_0(K)$ has the (uniform) BCP if and only if $K$ has a countable $π$-basis. Moreover, we give the stability theorem: the vector-valued continuous function space $C_0(K,X)$ has the (strong or uniform) BCP if and only if $K$ has a countable $π$-bas…
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 $L_p$.
We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.
Given $1< p < q < \infty$ it is well know that the natural embedding of Lebesgue sequence spaces $\ell_p \hookrightarrow \ell_q$ is strictly singular. In this paper we extend this classical results and show that even the natural non-compact embedding between Lebesgue and weak Lebesgue sequence spaces $\ell_{p}\hookrightarrow \ell_{p,\infty}$ is strictly singular.
We give sufficient conditions on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in $\mathcal{L}(X)$, the space of all operators on $X$. We say that a basic sequence $(e_n)$ is quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, we have that $(e_{k_n})$ dominates $(e_{\ell_n})$. We prove that if a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in $\mathcal{L}(X)$.
Let $E$ be a Banach space that does not contain any copy of $\ell^1$ and $\A$ be a non commutative $C^*$-algebra. We prove that every absolutely summing operator from $\A$ into $E^*$ is compact, thus answering a question of Pełczynski. As application, we show that if $G$ is a compact metrizable abelian group and $Λ$ is a Riesz subset of its dual then every countably additive $\A^*$-valued measure with bounded variation and whose Fourier transform is supported by $Λ$ has relatively compact range. Extensions of the same result to symmetric spaces of measurable operators are also presented.
We study nuclear embeddings for spaces of Besov and Triebel-Lizorkin type defined on quasi-bounded domains $Ω\subset {\mathbb R}^d$. The counterpart for such function spaces defined on bounded domains has been considered for a long time and the complete answer was obtained only recently. Compact embeddings for function spaces defined on quasi-bounded domains have been studied in detail already, also concerning their entropy and $s$-numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has been introduce by Grothendieck in 1955 already. Our se…
We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of $\ell_p^n$, $p > 0$, into various type of Banach or quasi-Banach spaces. In particular, for $0 <r < p<2$ with $r \le 1$, we construct a family of operators that embed $\ell_p^n$ into $\ell_r^{(1+η)n}$, with optimal polynomial bounds in $η>0$.
We study the embedding $\text{id}: \ell_p^b(\ell_q^d) \to \ell_r^b(\ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(\text{id})$ provided that $0<p<r\leq \infty$ and $0<q\leq u\leq \infty$. Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel-Lizorkin spaces of small dominating mixed smoothness which settles an open question in the literature. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.
In this paper we first show that if $X$ is a Banach space and $α$ is a left invariant crossnorm on $\ell_\infty\otimes X$, then there is a Banach lattice $L$ and an isometric embedding $J$ of $X$ into $L$, so that $I\otimes J$ becomes an isometry of $\ell_\infty\otimes_αX$ onto $\ell_\infty\otimes_m J(X)$. Here $I$ denotes the identity operator on $\ell_\infty$ and $\ell_\infty\otimes_m J(X)$ the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to characterize the Gordon-Lewis property $\GL$ in terms of emb…
Given $0<p,q, r<\infty $ and $ q<r\le \infty$ we consider the natural embedding $\ell_{p,q}\hookrightarrow \ell_{p,r}$ between Lorenz sequence spaces. We prove that this non-compact embedding is always strictly singular but not finitely strictly singular.
We give an elementary proof that the $H^p$ spaces over the unit disc (or the upper half plane) are the interpolation spaces for the real method of interpolation between $H^1$ and $H^\infty$. This was originally proved by Peter Jones. The proof uses only the boundedness of the Hilbert transform and the classical factorisation of a function in $H^p$ as a product of two functions in $H^q$ and $H^r$ with $1/q+1/r=1/p$. This proof extends without any real extra difficulty to the non-commutative setting and to several Banach space valued extensions of $H^p$ spaces. In particular, this proof easily e…
We study finite subsets of $\ell_p$ and show that, up to nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains the spaces $\ell_p^n$, $n \in \mathbb{N}$.
We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.
We prove that, for $1 \le p<q<\infty$, the partially ordered set $P(ω)/{\rm Fin}$ can be embedded into Borel equivalence relations between $\mathbb{R}^ω/\ell_p$ and $\mathbb{R}^ω/\ell_q$. Since there is an antichain of size continuum in $P(ω)/{\rm Fin}$, therefore there are continuum many incomparable Borel equivalence relations between $\mathbb{R}^ω/\ell_p$ and $\mathbb{R}^ω/\ell_q$.
It is known that if finite subsets of a locally finite metric space $M$ admit $C$-bilipschitz embeddings into $\ell_p$ $(1\le p\le \infty)$, then for every $ε>0$, the space $M$ admits a $(C+ε)$-bilipschitz embedding into $\ell_p$. The goal of this paper is to show that for $p\ne 2,\infty$ this result is sharp in the sense that $ε$ cannot be dropped out of its statement.
We study finite subsets of $\ell_2$, and more generally any metric space, and consider whether these isometrically embed into a Banach space. Our results partially answer a question of Ostrovskii, on whether every infinite-dimensional Banach space contains every finite subset of $\ell_2$ isometrically. The updated version contains acknowledgement that Theorem 3.1 has been proven previously in a paper of Shkarin.
In this paper we prove that $L_{p}$ does not admit an equivariant coarse embedding into $\ell_p$ i.e there is no proper, affine, isometric action of $L_{p}$, viewed as a group under addition with the standard metric $|| . ||_p$, on $\ell_p$. This is done by showing that representations of $L_{p}$ into $ Isom(\ell_p)$ has to be trivial, which allows us to reduce the question to bi-Lipschitz setting.
In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from $\ell_q$ into $\ell_p$ ($0<p<q\le 1$) and new insights on the coarse embedabbility problem from $L_q$ into $\ell_q$, $q>2$. Relevant to geometric group theory purposes, the exact $\ell_p$-compressions of $\ell_2$ are computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.
We consider the problem of determining the asymptotic order of the Gelfand numbers of mixed-(quasi-)norm embeddings $\ell^b_p(\ell^d_q) \hookrightarrow \ell^b_r(\ell^d_u)$ given that $p \leq r$ and $q \leq u$, with emphasis on cases with $p\leq 1$ and/or $q\leq 1$. These cases turn out to be related to structured sparsity. We obtain sharp bounds in a number of interesting parameter constellations. Our new matching bounds for the Gelfand numbers of the embeddings of $\ell_1^b(\ell_2^d)$ and $\ell_2^b(\ell_1^d)$ into $\ell_2^b(\ell_2^d)$ imply optimality assertions for the recovery of block-spar…
$\newcommand{\eps}{\varepsilon} $We prove that for any $\eps > 0$ it is $\textsf{NP}$-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight $\textsf{NP}$-hardness result for the commutative Little Grothendi…
