k-Extreme Points in Symmetric Spaces of Measurable Operators
Anna Kamińska2014, Integral Equations and Operator Theory
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ and E be a strongly symmetric Banach function space on [0, τ (1)). We show that an operator x in the unit sphere of E(M, τ ) is k-extreme, k ∈ N, whenever its singular value function µ(x) is k-extreme and one of the following conditions hold (i) µ(∞, x) = limt→∞ µ(t, x) = 0 or (ii) n(x)Mn(x * ) = 0 and |x| ≥ µ(∞, x)s(x), where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever M is nonatomic. The global k-rotundity property follows, that is if M is non-atomic then E is k-rotund if and only if E(M, τ ) is k-rotund. As a consequence of the noncommutive results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement µ(f ) is k-extreme and |f | ≥ µ(∞, f ). We conclude with the corollary on orbits Ω(g) and Ω ′ (g). We get that f is a k-extreme point of the orbit Ω(g), g ∈ L 1 + L∞, or Ω ′ (g), g ∈ L 1 [0, α), α < ∞, if and only if µ(f ) = µ(g) and |f | ≥ µ(∞, f ). From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.
Related papers
Note on extreme points in Marcinkiewicz function spaces
Banach Journal of Mathematical Analysis, 2010
We show that the unit ball of the subspace M 0 W of ordered continuous elements of M W has no extreme points, where M W is the Marcinkiewicz function space generated by a decreasing weight function w over the interval (0, ∞) and W (t) = t 0 w, t ∈ (0, ∞). We also present here a proof of the fact that a function f in the unit ball of M W is an extreme point if and only if f * = w.
Extremity in Köthe–Bochner Function Spaces
Journal of Mathematical Analysis and Applications, 1998
Exposed and strongly exposed points in symmetric spaces of measurable operators
Houston Journal of Mathematics, 2013
We investigate the relationships between exposed or strongly exposed points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M, τ) associated to a semifinite von Neumann algebra (M, τ). We prove that an operator x is an exposed or strongly exposed point of the unit ball of the symmetric space of measurable operators E(M, τ) if and only if its singular value function µ(x) is an exposed or strongly exposed point of the unit ball in E, respectively. 1991 Mathematics Subject Classification. 46B20, 46B28, 47L05, 47L20. Key words and phrases. Symmetric spaces of measurable operators, unitary matrix spaces, exposed points, strongly exposed point. s 0 µ(t; f) dt s 0 µ(t; g) dt for every s > 0, then we say that f is submajorized by g and write f ≺ g. It is said that E is fully symmetric whenever for g ∈ E and f ∈ L 0 [0, α), if f ≺ g then f ∈ E and f E g E. If E is order continuous then it is fully symmetric [17, Chapter II, Theorem 4.10]. We refer to monographs [2, 17] for theory of Banach function spaces. Let us recall some basic facts regarding theory of operator algebras. We denote by M a semifinite von Neumann algebra, on the complex Hilbert space H, with a faithful, normal, semifinite trace τ and unit element 1. If x ∈ M then by x M we mean the norm of x in the space of all bounded linear operators B(H). Any closed and densely defined operator x : D(x) → H with domain D(x) ⊂ H is τ-measurable (with respect to the von Neumann algebra M), if ux = xu for all unitary operators u in the commutant M of M and there exists λ > 0 such that τ e |x| (λ, ∞) < ∞, where e |x| (•) is a spectral measure of the operator |x|. By S (M, τ) we denote the collection of all τ-measurable operators. S (M, τ) is a *-algebra with respect to the strong sum, the strong product, and the adjoint operation. Given , δ > 0, let U (, δ) = x ∈ S (M, τ) : xp M , τ (p ⊥) δ for some p ∈ P(M) , where P(M) is the subalgebra of all orthogonal projections in M and p ⊥ = 1 − p. The sets {U (, δ)} form a base at 0 for a metrizable Hausdorff topology, which is called the measure topology. Convergence with respect to this topology is denoted by x n τ → x, x n , x ∈ S (M, τ). The following idea of submajorization is useful and applied extensively in the sequel. For an operator x ∈ S (M, τ), the function µ(t; x) defined by µ(t; x) = inf s 0 : τ (e |x| (s, ∞)) t , t 0,
Extreme Points of the Unit Ball in the Dual Space of Some Real Subspaces of Banach Spaces of Lipschitz Functions
Let X be a compact Hausdorff space, τ be a continuous involution on X and C X, τ denote the uniformly closed real subalgebra of C X consisting of all f ∈ C X for which f • τ f. Let X, d be a compact metric space and let Lip X, d α denote the complex Banach space of complexvalued Lipschitz functions of order α on X, d under the norm f X,pα max{ f X , p α f }, where α ∈ 0, 1. For α ∈ 0, 1 , the closed subalgebra of Lip X, α consisting of all f ∈ Lip X, d α for which |f x − f y |/d α x, y → 0 as d x, y → 0, denotes by lip X, d α. Let τ be a Lipschitz involution on X, d and define Lip X, τ, d α Lip X, d α ∩ C X, τ for α ∈ 0, 1 and lip X, τ, d α lip X, d α ∩ C X, τ for α ∈ 0, 1. In this paper, we give a characterization of extreme points of B A * , where A is a real linear subspace of Lip X, d α or lip X, d α which contains 1, in particular, Lip X, τ, d α or lip X, τ, d α .
A note on extreme points in dual spaces
Acta Mathematica Sinica, English Series, 2013
Given a normed space X it can be easily proven that every extreme point in B X * , the unit ball of X * , is the restriction of an extreme point in B X * * *. Our purpose is to study when the restrictions of extreme points in B X * * * are extreme points in B X *. Namely, we characterize L 1-preduals satisfying the aforementioned property.
Extreme points of the unit ball of an operator space
2004
This paper is a revision and an enlargement of the previous version titled "Extreme points of the unit ball of a quasi-multiplier space" which had been circulated since 2004. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) "quasi-identities". Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a $C^*$-algebra or a one-sided ideal in a $C^*$-algebra in terms of quasi-multipliers. An extreme point is also used to show that any TRO with predual can be decomposed to the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra.
On Extreme Points of Subordination Families
Proceedings of the American Mathematical Society, 1983
Let F be the set of analytic functions in (/-(r: |; |< 1} subordinate to a univalent function/. Let D = f(U). For g(z) = f(<t>(:)) E F. let \{0) denote the distance between g(e'e) and dD (boundary of D) We obtain the following results. (l)If/'isNevanlinnathen/"2*logA(0)i/0 =-oo if and only if /2"log(l-| *(,'*) |)di=-00. (2) If g is an extreme point of the closed convex hull of F then rZ-rr for the case when D is a Jordan domain subset to a half-plane and/' is Nevanlinna /2"log(l-\<t>(e'")\)de =-oo. 1. Introduction. Let U-[z: \z\< 1} and let A denote the set of functions analytic in U. Let B0 denote the subset of A consisting of functions <j> that satisfy | <p(z) |< 1 for z E U and <H0) = 0. Throughout this paper we assume that f E A and / is univalent in U. Let F denote the subset of A consisting of functions g that are subordinate to / in U. This means that g E A, g(0) =/(0) and g(U) Cf(U). These conditions are equivalent to the existence oí 4> E B0 so that g(z)-/(<p(z)). F is characterized by (D g(z)=f(<t>(z)) where <f> E B0. Equation (1) defines a one-to-one correspondence between F and B0. Let D denote/(t/). For g E F, let (2) g{y') = limg(re"). r~\ Since f E Hp, for p<{, g(e) exists almost everywhere. Let X(6) denote the distance between g(e'9) and dD where 9Z> denotes the boundary of D. T. H. MacGregor and the author [1] proved that if / is convex, bounded, and if dD is sufficiently smooth, then g is an extreme point of F if and only if (3) (2"\og\(0)d0 Jet-00. This result implies the well-known fact that <p is an extreme point of B0 if and only if (4) r2wlog(l-| *(e'*) |) £/» =-oo" •'n
Extreme points of the unit ball of a quasi-multiplier space
We study extreme points of the unit ball of the set of quasi-multipliers of an operator space by introducing the new notion: (approximate) quasi-identities. We give a necessary and sufficient condition for an operator space to become an operator algebra with a contractive approximate quasi-(respectively, left, right, two-sided) identity in terms of extreme points of contractive quasi-multipliers. We also give a necessary sufficient condition for an operator space to become a C * -algebra. Furthermore, we answer the open question about Properties (L) and (R) raised by D. P. Blecher.
A class of extreme contractions inL(l p )
Annali di Matematica Pura ed Applicata, 1990
In this note we prove that: i] T is a contraction in L(lP) that maps elements o/ disjoint support to elements o/ disjoint support, then T is an extreme point o] the unit ball o] L(l~), l < p < c% pv~ 2, i] and only i] T is o] the ]orm T = ~. ~i | Y~, where i=l either (Yi) ]orm a p.orthonormal sequence or the nonzero elements o/(Yi) ]orm a p-orthenormal sequence /or which U supp (y~) = h r.
Preservation of Extreme Points
Mathematics
We characterize the extreme points of the closed unit ball of the dual of a Banach space which are preserved by the adjoint of any extreme operator. The result is related to the structure topology introduced by Alfsen and Effros on the set of all extreme points in the dual of any Banach space. As a consequence, we prove that c0(I) is the only Banach space such that the adjoint of every extreme operator taking values into it preserves extreme points.
Quasi-multipliers of an operator space. II. Extreme points and quasi-identities
Journal of Operator Theory, 2016
We study extreme points of the unit ball of an operator space by introducing the new notion “(approximate) quasi-identities”. More specifically, we characterize an operator algebra having a contractive (approximate) quasi– (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points of the unit ball (of the weak*-closure) of the underlying operator space. Furthermore, we give a necessary and sufficient condition for a given operator space to be qualified to become a C*-algebra or a one-sided ideal in a C*-algebra in terms of quasi-multipliers.
Extremal Unital Completely Positive Maps and Their Symmetries
Complex Analysis and Operator Theory, 2018
We consider the set P 1 (A, M) (respectively C P 1 (A, M) of unital positive (completely) maps from a C * algebra A to a von-Neumann sub-algebra M of B(H), the algebra of bounded linear operators on a Hilbert space H. We study the extreme points of the convex set P 1 (A, M) (C P 1 (A, M)) via their canonical lifting to the convex set of (unital) positive (completely) normal maps from to M, where A * * is the universal enveloping von-Neumann algebra over A. If A = M then a (completely) positive map τ admits a unique decomposition into a sum of a normal and a singular (completely) positive maps. Furthermore, if M is a factor then a unital complete positive map is a unique convex combination of unital normal and singular completely positive maps. We also used a duality argument to find a criteria for an element in the convex set of unital completely positive maps with a given faithful normal invariant state on M to be extremal. In our investigation, gauge symmetry in the minimal Stinespring representation of a completely positive map and Kadison theorem on order isomorphism played an important role. Keywords Operator system • Arveson-Hahn-Banach extension theorem • Complete order isomorphism Mathematics Subject Classification 46L Communicated by Daniel Aron Alpay.
Subordination families and extreme points
Transactions of the American Mathematical Society, 1988
Let s ( F ) s(F) denote the set of functions subordinate to a univalent function F F in Δ \Delta the unit disk. Let B 0 {B_0} denote the set of functions ϕ ( z ) \phi (z) analytic in Δ \Delta satisfying | ϕ ( z ) | > 1 |\phi (z)| > 1 and ϕ ( 0 ) = 0 \phi (0) = 0 . We prove that if f = F ∘ ϕ f = F \circ \phi is an extreme point of s ( F ) s(F) , then ϕ \phi is an extreme point of B 0 {B_0} . Let D = F ( s ) D = F(s) and λ ( w , ∂ D ) \lambda (w,\,\partial D) denote the distance between w w and ∂ D \partial D (boundary of D D ). We also prove that if ϕ \phi is an extreme point of B 0 {B_0} and | ϕ ( e i t ) | > 1 |\phi ({e^{it}})| > 1 for almost all t t , then ∫ 0 2 π log λ ( F ( ϕ ( e i t ) e i θ ) , ∂ D ) d t = − ∞ \int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}),\,\partial D)\,dt = - \infty } for almost all θ \theta .
Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators
Studia Mathematica, 2010
We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(M, τ) of τ-measurable operators associated to a semifinite von Neumann algebra (M, τ) or of the unit ball in the unitary matrix space CE. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric space E(M, τ) inherit these properties from their singular value function µ(x) in the unit ball of E with additional necessary requirements on x in the case of complex extreme points. We also obtain the full converse statements for the von Neumann algebra M with a faithful, normal, σ-finite trace τ as well as for the unitary matrix space CE. Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.
Weakly extremal properties of Banach spaces
Mathematical Notes of the Academy of Sciences of the USSR, 1989
Mean ergodicity of positive operators in KB-spaces
Journal of Mathematical Analysis and Applications, 2006
Endpoint Mapping Properties of Spherical Maximal Operators
Journal of the Institute of Mathematics of Jussieu, 2003
Extreme points and isometries of φ-nuclear operators in Hilbert spaces
Journal of Mathematical Analysis and Applications, 1990
A modulus function 4 is a continuous strictly increasing subadditive real valued function 4: [0, co) + [0, co) for which d(O) = 0. The object of this paper is to define d-nuclear operators in Banach spaces. The basic properties of these operators are studied. In particular it is proved that &nuclear operators are stable under injective tensor product. In case of Hilbert spaces, the extreme points of the unit ball and the isomerries of such class of operators are characterized. $2
Maximal Operators and Characterization of Hardy Spaces
Analysis Mathematica, 2020
It is known that the maximal operator σ * f is of type (H p , L p ) if the Vilenkin group G is bounded and p > 1 2 . We prove a maximal converse inequality which characterizes the space H p by means of the operator σ † f := sup n |σM n f |, for bounded groups.
A note on the minimum property
Proceedings of the American Mathematical Society, 1988
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.