timur oikhberg | University of Illinois at Urbana-Champaign (original) (raw)

Papers by timur oikhberg

Mathematische Zeitschrift, Jul 1, 2023

Mathematische Annalen

In this article, we construct various subspaces doing stable phase retrieval, and make connection... more In this article, we construct various subspaces doing stable phase retrieval, and make connections with Lambda(p)\Lambda(p)Lambda(p)-set theory. Moreover, we set the foundations for an analysis of stable phase retrieval in general function spaces. This, in particular, allows us to show that Hölder stable phase retrieval implies stable phase retrieval, improving the stability bounds in a recent article of M. Christ and the third and fourth authors. We also characterize those compact Hausdorff spaces KKK such that C(K)C(K)C(K) contains an infinite dimensional SPR subspace.

arXiv (Cornell University), Apr 12, 2023

We continue the study initiated in [6] of properties related to greedy bases in the case when the... more We continue the study initiated in [6] of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from [2]. In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

arXiv (Cornell University), Mar 9, 2023

We begin by describing the unit ball of the free p-convex Banach lattice over a Banach space E (d... more We begin by describing the unit ball of the free p-convex Banach lattice over a Banach space E (denoted by FBL (p) [E]) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space E has the λ-Approximation Property, then FBL (p) [E] has the λ-Positive Approximation Property. Further, we show that operators u ∈ B(E, F) (where E and F are Banach spaces) which extend to lattice homomorphisms from FBL (q) [E] to FBL (p) [F ] are precisely those whose adjoints are (q, p)-mixing. Related results are also obtained for free lattices with an upper p-estimate. Contents 1. Introduction and preliminaries 2 2. Representing unit balls as solid convex hulls 4 2.1. Survey of solid convex hulls 4 2.2. The unit ball of FBL (p) [E] 4 2.3. The unit ball of FBL ↑p [E] 9 3. Positive Bounded Approximation Property 10 4. Duals of free Banach lattices 13 4.1. Sums of atoms in FBL ↑p [E] * 13 4.2. Sums of atoms in FBL (p) [E] * 14

De Gruyter eBooks, Aug 10, 2020

We consider the "order" analogues of some classical notions of Banach space geometry: extreme poi... more We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally, we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodým Property.

arXiv (Cornell University), Oct 8, 2016

We study the stability of band preserving operators on Banach lattices. To this end the notion of... more We study the stability of band preserving operators on Banach lattices. To this end the notion of ε-band preserving mapping is introduced. It is shown that, under quite general assumptions, a ε-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of ε-band preserving maps are also obtained.

arXiv (Cornell University), Sep 24, 2002

M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant ... more M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T * has an eigenvector. We present generalizations of this result as well as some applications to C *-algebras, operators on ℓ 1 , operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras. M. Krein proved in [KR48, Theorem 3.3] that if T is a continuous operator on a normed space leaving invariant a non-empty open cone, then its adjoint T * has an eigenvector. Krein's result has an immediate application to the Invariant Subspace Problem because of the following observation. If T is a bounded operator on a Banach space and not a multiple of the identity, and T * f = λf , then the kernel of f is a closed non-trivial subspace of codimension 1 which is invariant under T. Moreover, Range(λI − T) is a closed nontrivial subspace which is proper (it is contained in the kernel of f) and hyperinvariant for T , that is, it is invariant under every operator commuting with T. Several proofs and modifications of Krein's theorem appear in the literature, see, e.g., [AAB92, Theorems 6.3 and 7.1] and [S99, p. 315]. We prove yet another version of Krein's Theorem: if T is a positive operator on an ordered normed space in which the unit ball has a dominating point, then T * has a positive eigenvector. We deduce the original Krein's version of the theorem from this, as well as several applications and related results. In particular, we show that if a bounded operator T on a Banach space satisfies any of the following conditions, then T * has an eigenvector. Moreover, if the condition holds for a commutative family of operators, then the family of the adjoint operators has a common eigenvector.

arXiv (Cornell University), Jan 17, 2007

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented... more In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L p-space, then it is either a L p-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non commutative L p-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence σ we prove that most of these spaces are operator L p-spaces, not completely isomorphic to previously known such spaces. However it turns out that some column and row versions of our spaces are not operator L p-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.

arXiv (Cornell University), Sep 22, 2010

For an operator TinB(X,Y)T \in B(X,Y)TinB(X,Y), we denote by am(T)a_m(T)am(T), cm(T)c_m(T)cm(T), dm(T)d_m(T)dm(T), and tm(T)t_m(T)tm(T) its appro... more For an operator TinB(X,Y)T \in B(X,Y)TinB(X,Y), we denote by am(T)a_m(T)am(T), cm(T)c_m(T)cm(T), dm(T)d_m(T)dm(T), and tm(T)t_m(T)tm(T) its approximation, Gelfand, Kolmogorov, and absolute numbers. We show that, for any infinite dimensional Banach spaces XXX and YYY, and any sequence alphamsearrow0\alpha_m \searrow 0alphamsearrow0, there exists TinB(X,Y)T \in B(X,Y)TinB(X,Y) for which the inequality 3alphalceilm/6rceilgeqam(T)geqmaxcm(t),dm(T)geqmincm(t),dm(T)geqtm(T)geqalpham/93 \alpha_{\lceil m/6 \rceil} \geq a_m(T) \geq \max\{c_m(t), d_m(T)\} \geq \min\{c_m(t), d_m(T)\} \geq t_m(T) \geq \alpha_m/93alphalceilm/6rceilgeqam(T)geqmaxcm(t),dm(T)geqmincm(t),dm(T)geqtm(T)geqalpham/9 holds for every minNm \in \NminN. Similar results are obtained for other sss-scales.

arXiv (Cornell University), Nov 21, 2022

The nonlinear geometry of operator spaces has recently started to be investigated. Many notions o... more The nonlinear geometry of operator spaces has recently started to be investigated. Many notions of nonlinear embeddability have been introduced so far, but, as noticed before by other authors, it was not clear whether they could be considered "correct notions". The main goal of these notes is to provide the missing evidence to support that almost complete coarse embeddability is "a correct notion". This is done by proving results about the complete isomorphic theory of ℓ 1-sums of certain operators spaces. Several results on the complete isomorphic theory of c 0-sums of operator spaces are also obtained.

arXiv (Cornell University), Jul 24, 2017

We find large classes of injective and projective p-multinormed spaces. In fact, these classes ar... more We find large classes of injective and projective p-multinormed spaces. In fact, these classes are universal, in the sense that every pmultinormed space embeds into (is a quotient of) an injective (resp. projective) p-multinormed space. As a consequence, we show that any pmultinormed space has a canonical representation as a subspace of a quotient of a Banach lattice.

arXiv (Cornell University), Apr 3, 1997

In analogy with the maximal tensor product of C *-algebras, we define the "maximal" tensor produc... more In analogy with the maximal tensor product of C *-algebras, we define the "maximal" tensor product E 1 ⊗ µ E 2 of two operator spaces E 1 and E 2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor

arXiv (Cornell University), Jul 4, 2012

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholdin... more We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best Nterm error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic bases. We define the quasi-greedy constant K of the basis B to be the least K such that (1.2) holds for all permutations π satisfying (1.1).

arXiv (Cornell University), Nov 10, 2018

We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algo... more We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.

arXiv (Cornell University), Sep 28, 2010

In their previous paper [4], the authors investigated the existence of an element x of a quasi-Ba... more In their previous paper [4], the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (A n) (defined by E(x, A n) = inf a∈An x − a n) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive.

Journal of Mathematical Analysis and Applications, Sep 1, 2011

Elements a and b of a C *-algebra are called orthogonal (a ⊥ b) if a * b = ab * = 0. We say that ... more Elements a and b of a C *-algebra are called orthogonal (a ⊥ b) if a * b = ab * = 0. We say that vectors x and y in a Banach space X are semi-M-orthogonal (x ⊥ SM y) if x ± y max{ x , y }. We prove that every linear bijection T : A → X, where X is a Banach space, A is either a von Neumann algebra or a compact C *-algebra, and T (a) ⊥ SM T (b) whenever a ⊥ b, must be continuous. Consequently, every complete (semi-)M-norm on a von Neumann algebra or on a compact C *-algebra is automatically continuous.

Rocky Mountain Journal of Mathematics, Apr 1, 2007

It is proved that ρ is induced by a norm provided it is translation invariant, real scalar "separ... more It is proved that ρ is induced by a norm provided it is translation invariant, real scalar "separately" continuous, such that every 1-dimensional subspace of X is isometric to K in its natural metric, and (in the complex case) ρ(x, y) = ρ(ix, iy) for any x, y ∈ X.

Journal of Approximation Theory, Apr 1, 1995

The absolute widths of the natural identity operator I n pq , mapping l n p to l n q , for 1 ≤ q ... more The absolute widths of the natural identity operator I n pq , mapping l n p to l n q , for 1 ≤ q ≤ p ≤ ∞ are considered. The main result of this work is the following Theorem 1 If 1 ≤ q ≤ p ≤ ∞, 0 ≤ m ≤ n, d a m (I n pq) = (n − m) 1/q−1/p where d a m (u) stands for the absolute width of the operator u .

Revista Matematica Complutense, Nov 18, 2011

Elements a and b of a non-commutative L p (M, τ) space associated to a von Neumann algebra, M, eq... more Elements a and b of a non-commutative L p (M, τ) space associated to a von Neumann algebra, M, equipped with a normal semi-finite faithful trace τ , are called orthogonal if l(a)l(b) = r(a)r(b) = 0, where l(x) and r(x) denote the left and right support projections of x. A linear map T from L p (M, τ) to a normed space X is said to be orthogonality-top -orthogonality preserving if T (a) + T (b) p = a p + b p whenever a and b are orthogonal. In this paper, we prove that an orthogonalityto-p-orthogonality preserving linear bijection from L p (M, τ) to a Banach space is automatically continuous if 1 ≤ p < ∞, and M is either an abelian von Neumann algebra or a discrete von Neumann algebras. Furthermore, any complete p-additive norm on such L p (M, τ) is equivalent to the canonical norm. Keywords Non-commutative L p spaces • Banach lattices • Von Neumann algebras • Orthogonality preservers • p-orthogonality Mathematics Subject Classification (2000) 46B04 • 46B42 • 46L52

arXiv (Cornell University), Oct 2, 2022

Mathematische Zeitschrift, Jul 1, 2023

Mathematische Annalen

In this article, we construct various subspaces doing stable phase retrieval, and make connection... more In this article, we construct various subspaces doing stable phase retrieval, and make connections with Lambda(p)\Lambda(p)Lambda(p)-set theory. Moreover, we set the foundations for an analysis of stable phase retrieval in general function spaces. This, in particular, allows us to show that Hölder stable phase retrieval implies stable phase retrieval, improving the stability bounds in a recent article of M. Christ and the third and fourth authors. We also characterize those compact Hausdorff spaces KKK such that C(K)C(K)C(K) contains an infinite dimensional SPR subspace.

arXiv (Cornell University), Apr 12, 2023

We continue the study initiated in [6] of properties related to greedy bases in the case when the... more We continue the study initiated in [6] of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from [2]. In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

arXiv (Cornell University), Mar 9, 2023

We begin by describing the unit ball of the free p-convex Banach lattice over a Banach space E (d... more We begin by describing the unit ball of the free p-convex Banach lattice over a Banach space E (denoted by FBL (p) [E]) as a closed solid convex hull of an appropriate set. Based on it, we show that, if a Banach space E has the λ-Approximation Property, then FBL (p) [E] has the λ-Positive Approximation Property. Further, we show that operators u ∈ B(E, F) (where E and F are Banach spaces) which extend to lattice homomorphisms from FBL (q) [E] to FBL (p) [F ] are precisely those whose adjoints are (q, p)-mixing. Related results are also obtained for free lattices with an upper p-estimate. Contents 1. Introduction and preliminaries 2 2. Representing unit balls as solid convex hulls 4 2.1. Survey of solid convex hulls 4 2.2. The unit ball of FBL (p) [E] 4 2.3. The unit ball of FBL ↑p [E] 9 3. Positive Bounded Approximation Property 10 4. Duals of free Banach lattices 13 4.1. Sums of atoms in FBL ↑p [E] * 13 4.2. Sums of atoms in FBL (p) [E] * 14

De Gruyter eBooks, Aug 10, 2020

We consider the "order" analogues of some classical notions of Banach space geometry: extreme poi... more We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally, we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodým Property.

arXiv (Cornell University), Oct 8, 2016

We study the stability of band preserving operators on Banach lattices. To this end the notion of... more We study the stability of band preserving operators on Banach lattices. To this end the notion of ε-band preserving mapping is introduced. It is shown that, under quite general assumptions, a ε-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of ε-band preserving maps are also obtained.

arXiv (Cornell University), Sep 24, 2002

M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant ... more M. Krein proved in [KR48] that if T is a continuous operator on a normed space leaving invariant an open cone, then its adjoint T * has an eigenvector. We present generalizations of this result as well as some applications to C *-algebras, operators on ℓ 1 , operators with invariant sets, contractions on Banach lattices, the Invariant Subspace Problem, and von Neumann algebras. M. Krein proved in [KR48, Theorem 3.3] that if T is a continuous operator on a normed space leaving invariant a non-empty open cone, then its adjoint T * has an eigenvector. Krein's result has an immediate application to the Invariant Subspace Problem because of the following observation. If T is a bounded operator on a Banach space and not a multiple of the identity, and T * f = λf , then the kernel of f is a closed non-trivial subspace of codimension 1 which is invariant under T. Moreover, Range(λI − T) is a closed nontrivial subspace which is proper (it is contained in the kernel of f) and hyperinvariant for T , that is, it is invariant under every operator commuting with T. Several proofs and modifications of Krein's theorem appear in the literature, see, e.g., [AAB92, Theorems 6.3 and 7.1] and [S99, p. 315]. We prove yet another version of Krein's Theorem: if T is a positive operator on an ordered normed space in which the unit ball has a dominating point, then T * has a positive eigenvector. We deduce the original Krein's version of the theorem from this, as well as several applications and related results. In particular, we show that if a bounded operator T on a Banach space satisfies any of the following conditions, then T * has an eigenvector. Moreover, if the condition holds for a commutative family of operators, then the family of the adjoint operators has a common eigenvector.

arXiv (Cornell University), Jan 17, 2007

In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented... more In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an L p-space, then it is either a L p-space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non commutative L p-spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which can be considered as operator space analogues of the Rosenthal sequence spaces from Banach space theory, constructed in 1970. Under the usual conditions on the defining sequence σ we prove that most of these spaces are operator L p-spaces, not completely isomorphic to previously known such spaces. However it turns out that some column and row versions of our spaces are not operator L p-spaces and have a rather complicated local structure which implies that the Lindenstrauss-Rosenthal alternative does not carry over to the non-commutative case.

arXiv (Cornell University), Sep 22, 2010

For an operator TinB(X,Y)T \in B(X,Y)TinB(X,Y), we denote by am(T)a_m(T)am(T), cm(T)c_m(T)cm(T), dm(T)d_m(T)dm(T), and tm(T)t_m(T)tm(T) its appro... more For an operator TinB(X,Y)T \in B(X,Y)TinB(X,Y), we denote by am(T)a_m(T)am(T), cm(T)c_m(T)cm(T), dm(T)d_m(T)dm(T), and tm(T)t_m(T)tm(T) its approximation, Gelfand, Kolmogorov, and absolute numbers. We show that, for any infinite dimensional Banach spaces XXX and YYY, and any sequence alphamsearrow0\alpha_m \searrow 0alphamsearrow0, there exists TinB(X,Y)T \in B(X,Y)TinB(X,Y) for which the inequality 3alphalceilm/6rceilgeqam(T)geqmaxcm(t),dm(T)geqmincm(t),dm(T)geqtm(T)geqalpham/93 \alpha_{\lceil m/6 \rceil} \geq a_m(T) \geq \max\{c_m(t), d_m(T)\} \geq \min\{c_m(t), d_m(T)\} \geq t_m(T) \geq \alpha_m/93alphalceilm/6rceilgeqam(T)geqmaxcm(t),dm(T)geqmincm(t),dm(T)geqtm(T)geqalpham/9 holds for every minNm \in \NminN. Similar results are obtained for other sss-scales.

arXiv (Cornell University), Nov 21, 2022

The nonlinear geometry of operator spaces has recently started to be investigated. Many notions o... more The nonlinear geometry of operator spaces has recently started to be investigated. Many notions of nonlinear embeddability have been introduced so far, but, as noticed before by other authors, it was not clear whether they could be considered "correct notions". The main goal of these notes is to provide the missing evidence to support that almost complete coarse embeddability is "a correct notion". This is done by proving results about the complete isomorphic theory of ℓ 1-sums of certain operators spaces. Several results on the complete isomorphic theory of c 0-sums of operator spaces are also obtained.

arXiv (Cornell University), Jul 24, 2017

We find large classes of injective and projective p-multinormed spaces. In fact, these classes ar... more We find large classes of injective and projective p-multinormed spaces. In fact, these classes are universal, in the sense that every pmultinormed space embeds into (is a quotient of) an injective (resp. projective) p-multinormed space. As a consequence, we show that any pmultinormed space has a canonical representation as a subspace of a quotient of a Banach lattice.

arXiv (Cornell University), Apr 3, 1997

In analogy with the maximal tensor product of C *-algebras, we define the "maximal" tensor produc... more In analogy with the maximal tensor product of C *-algebras, we define the "maximal" tensor product E 1 ⊗ µ E 2 of two operator spaces E 1 and E 2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor

arXiv (Cornell University), Jul 4, 2012

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholdin... more We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best Nterm error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic bases. We define the quasi-greedy constant K of the basis B to be the least K such that (1.2) holds for all permutations π satisfying (1.1).

arXiv (Cornell University), Nov 10, 2018

We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algo... more We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.

arXiv (Cornell University), Sep 28, 2010

In their previous paper [4], the authors investigated the existence of an element x of a quasi-Ba... more In their previous paper [4], the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (A n) (defined by E(x, A n) = inf a∈An x − a n) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive.

Journal of Mathematical Analysis and Applications, Sep 1, 2011

Elements a and b of a C *-algebra are called orthogonal (a ⊥ b) if a * b = ab * = 0. We say that ... more Elements a and b of a C *-algebra are called orthogonal (a ⊥ b) if a * b = ab * = 0. We say that vectors x and y in a Banach space X are semi-M-orthogonal (x ⊥ SM y) if x ± y max{ x , y }. We prove that every linear bijection T : A → X, where X is a Banach space, A is either a von Neumann algebra or a compact C *-algebra, and T (a) ⊥ SM T (b) whenever a ⊥ b, must be continuous. Consequently, every complete (semi-)M-norm on a von Neumann algebra or on a compact C *-algebra is automatically continuous.

Rocky Mountain Journal of Mathematics, Apr 1, 2007

It is proved that ρ is induced by a norm provided it is translation invariant, real scalar "separ... more It is proved that ρ is induced by a norm provided it is translation invariant, real scalar "separately" continuous, such that every 1-dimensional subspace of X is isometric to K in its natural metric, and (in the complex case) ρ(x, y) = ρ(ix, iy) for any x, y ∈ X.

Journal of Approximation Theory, Apr 1, 1995

The absolute widths of the natural identity operator I n pq , mapping l n p to l n q , for 1 ≤ q ... more The absolute widths of the natural identity operator I n pq , mapping l n p to l n q , for 1 ≤ q ≤ p ≤ ∞ are considered. The main result of this work is the following Theorem 1 If 1 ≤ q ≤ p ≤ ∞, 0 ≤ m ≤ n, d a m (I n pq) = (n − m) 1/q−1/p where d a m (u) stands for the absolute width of the operator u .

Revista Matematica Complutense, Nov 18, 2011

Elements a and b of a non-commutative L p (M, τ) space associated to a von Neumann algebra, M, eq... more Elements a and b of a non-commutative L p (M, τ) space associated to a von Neumann algebra, M, equipped with a normal semi-finite faithful trace τ , are called orthogonal if l(a)l(b) = r(a)r(b) = 0, where l(x) and r(x) denote the left and right support projections of x. A linear map T from L p (M, τ) to a normed space X is said to be orthogonality-top -orthogonality preserving if T (a) + T (b) p = a p + b p whenever a and b are orthogonal. In this paper, we prove that an orthogonalityto-p-orthogonality preserving linear bijection from L p (M, τ) to a Banach space is automatically continuous if 1 ≤ p < ∞, and M is either an abelian von Neumann algebra or a discrete von Neumann algebras. Furthermore, any complete p-additive norm on such L p (M, τ) is equivalent to the canonical norm. Keywords Non-commutative L p spaces • Banach lattices • Von Neumann algebras • Orthogonality preservers • p-orthogonality Mathematics Subject Classification (2000) 46B04 • 46B42 • 46L52

arXiv (Cornell University), Oct 2, 2022