Linear Maps on Factors which Preserve the Extreme Points of the Unit Ball

Mascioni, Vania; Molnár, Lajos

doi:10.4153/cmb-1998-057-7

Cited by 5 publications

(9 citation statements)

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“…So, φ is a linear map on B(H) which preserves the extreme points of the unit ball. The form of all linear maps with this property acting on a von Neumann factor was determined in [9]. The result Since our map φ is bijective, the same must hold for the corresponding morphism ψ or ψ ′ above.…”

Section: Resultsmentioning

confidence: 99%

Some linear preserver problems onB(H) concerning rank and corank

́r¹

1999

Linear Algebra and its Applications

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As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or finite corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a result on the form of linear maps which preserve the left ideals in B(H).1991 Mathematics Subject Classification. Primary: 47B49.

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Section: Resultsmentioning

confidence: 99%

Some linear preserver problems onB(H) concerning rank and corank

́r¹

1999

Linear Algebra and its Applications

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“…T (2, 1) = 3 2 v+ 1 2 w = [30]) are not expectable for general C * -algebras. We shall show that a more tractable description is possible for linear maps strongly preserving Brown-Pedersen quasi-invertibility.…”

Section: =======⇒mentioning

confidence: 99%

“…The other implications are, for the moment, unknown. We have already commented that V. Mascioni and L. Molnár characterized the linear maps on a von Neumann factor M preserving the extreme points of the unit ball of M in [30]. According to our terminology, they prove that, for a von Neumann factor M, a linear map T : M → M such that B(T (a), T (a)) = 0 whenever B(a, a) = 0, is a unital Jordan * -homomorphism multiplied by a unitary element (see [ However T (x ∧ ) may not coincide, in general, with T (x) ∧ .…”

Section: =======⇒mentioning

confidence: 99%

Linear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservers

Burgos¹,

Márquez-García²,

Campoy³

et al. 2016

Banach J. Math. Anal.

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We study new classes of linear preservers between C * -algebras and JB * -triples. Let E and F be JB * -triples with ∂ e (E 1 ). We prove that every linear map T : E → F strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C * -algebras A and B, for each linear map T strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan * -homomorphism S : A → B satisfying T (x) = T (1)S(x), for every x ∈ A. We also study the connections between linear maps strongly preserving Brown-Pedersen quasi-invertibility and other clases of linear preservers between C * -algebras like Bergmann-zero pairs preservers, Brown-Pedersen quasiinvertibility preservers and extreme points preservers.2010 Mathematics Subject Classification. 47B49, 15A09, 46L05, 47B48.between Banach algebras is a Jordan homomorphism if T (a 2 ) = T (a) 2 , for all a ∈ A (equivalently, T (ab + ba) = T (a)T (b) + T (b)T (a), for all a, b ∈ A). If A and B are unital, T is called unital if T (1) = 1. If A and B are C * -algebras and T (a * ) = T (a) * , for every a ∈ A, then T is called symmetric. Symmetric Jordan homomorphisms are named Jordan * -homomorphisms.Consequently, the problem of studying the linear maps T : A → B such that T (∂ e (A 1 )) ⊆ ∂ e (B 1 ) is a more general challenge, which is directly motivated by the just mentioned consequence of the Russo-Dye theorem. We only know partial answers to this problem. Concretely, V. Mascioni and L. Molnár studied the linear maps on a von Neumann factor M which preserve the extreme points of the unit ball of M. They prove that if M is infinite, then every linear mapping T on M preserving extreme points admits a factorization of the form T (a) = uS(a) (a ∈ M), where u is a (fixed) unitary in M and S either is a unital * -homomorphism or a unital * -anti-homomorphism (see [30, Theorem 1]). Theorem 2 in [30] states that, for a finite von Neumann algebra M, a linear map T : M → M preserves extreme points of the unit ball of M if and only if there exist a unitary operator u ∈ M and a unital Jordan * -homomorphism S : M → M such that T (a) = uS(a) (a ∈ A). In [29], L.E. Labuschagne and V.Mascioni study linear maps between C * -algebras whose adjoints preserve extreme points of the dual ball.The above results of Mascioni and L. Molnár are the most conclusive answers we know about linear maps between unital C * -algebras preserving extreme points. In this note we shall revisit the problem in full generality. We present several counter-examples to illustrate that the conclusions proved by Mascioni and Molnár for von Neumann factors need not be true for linear mappings preserving extreme points between unital C * -algebras (compare Remarks 5.8 and 5.9). It seems natural to ask whether a different class of linear preservers satisfies the same conclusions found by Mascioni and Molnár.Every unital Jordan homomorphism between Banach algebras strongly preserves invertibility, that is, T (a −1 ) = T (a) −1 , ...

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“…It is known that in general surjective isometries on the space of compact operators need not be given by composition by isometries of the underlying spaces (see [18]). See also [12] where extreme point-preserving surjections were described on L (H) again as compositions by unitaries or anti-unitaries. In particular, it follows from Corollary 1 in [12] that any nice operator on K (H) * whose adjoint is surjective, is weak * -continuous, a property which is well-known in the case of surjective isometries.…”

Section: Introductionmentioning

confidence: 99%

“…See also [12] where extreme point-preserving surjections were described on L (H) again as compositions by unitaries or anti-unitaries. In particular, it follows from Corollary 1 in [12] that any nice operator on K (H) * whose adjoint is surjective, is weak * -continuous, a property which is well-known in the case of surjective isometries. Motivated by these results in the first two sections of this paper we study nice surjections on K (X,Y ) that are of the form T → UTV for appropriate operators U and V .…”

Section: Introductionmentioning

confidence: 99%

Nice surjections on spaces of operators

Rao

2006

Proc. Indian Acad. Sci. (Math. Sci.)

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Abstract.A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni [9] of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on K (X,Y ) for Banach spaces X,Y . We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for L (X,Y ) thereby proving an analogue of the result from [9] for L p (1 < p = 2 < ∞) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from [8].

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Linear Maps on Factors which Preserve the Extreme Points of the Unit Ball

Cited by 5 publications

References 11 publications

Some linear preserver problems onB(H) concerning rank and corank

Some linear preserver problems onB(H) concerning rank and corank

Linear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservers

Nice surjections on spaces of operators

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