Linear disjointness preservers of operator algebras and related structures

Liu, Jung-Hui; Chou, Cheng‐Ying; Liao, Ching-Jou; Wong, Ngai–Ching

doi:10.14232/actasm-018-267-7

Cited by 21 publications

(12 citation statements)

References 70 publications

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“…Let us simply observe that a linear map T between Banach algebras is a homomorphism at zero if and only if it preserves zero products (i.e., ab = 0 implies T (a)T (b) = 0). We find in this way a natural link with the results on zero products preservers (see, for example, [1,2,8,10,28,29,32,33,[47][48][49][50][51] for additional details and results). Burgos, Cabello-Sánchez and the third author of this note explore in [6] those linear maps between C * -algebras which are * -homomorphisms at certain points of the domain, for example, at the unit element or at zero.…”

Section: Introductionsupporting

confidence: 69%

Linear Maps Which are Anti-derivable at Zero

Abulhamil

Jamjoom

Peralta

2020

Bull. Malays. Math. Sci. Soc.

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Let T : A → X be a bounded linear operator, where A is a C *-algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a) T is anti-derivable at zero (i.e., ab = 0 in A implies T (b)a + bT (a) = 0); (b) There exist an anti-derivation d : A → X * * and an element ξ ∈ X * * satisfying ξ a = aξ, ξ [a, b] = 0, T (ab) = bT (a) + T (b)a − bξ a, and T (a) = d(a) + ξ a, for all a, b ∈ A. We also prove a similar equivalence when X is replaced with A * *. This provides a complete characterization of those bounded linear maps from A into X or into A * * which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are *-anti-derivable at zero.

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Section: Introductionsupporting

confidence: 69%

Linear Maps Which are Anti-derivable at Zero

Abulhamil

Jamjoom

Peralta

2020

Bull. Malays. Math. Sci. Soc.

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“…Adjusting the proof of Theorem 2.1, we can achieve the equality U = V * , at the expenses that the diagonal matrices Q 1 , Q 2 may have negative entries. (7) If the domain is the set M n (C) of n × n complex matrices or the set H n (C) of n × n complex Hermitian matrices, our results can be deduced from the abstract theorems on C * -algebras; e.g., see [4,20,21,28], and also [6,27]. However, the proofs there do not seem to work for rectangular matrix spaces, or real square matrix spaces.…”

Section: Nonsurjective Preservers Of Disjointnessmentioning

confidence: 97%

“…The result is used to characterize linear maps that preserve the JB * -triple product, or just the zero triple product. Note that there are interesting results on disjointness preserving maps on different kinds of products over general operator spaces or algebras, see, e.g., [16,17,21,27,28]. However, the basic problem on disjointness preservers from a rectangular matrix space to another rectangular matrix space is unknown, and the existing results do not cover this case.…”

Section: Introductionmentioning

confidence: 99%

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Li¹,

Tsai²,

Wang³

et al. 2019

Preprint

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Let Mm,n be the space of m × n real or complex rectangular matrices. Two matrices A, B ∈ Mm,n are disjoint if A * B = 0n and AB * = 0m. In this paper, a characterization is given for linear maps Φ : Mm,n → Mr,s sending disjoint matrix pairs to disjoint matrix pairs, i.e., A, B ∈ Mm,n are disjoint ensures that Φ(A), Φ(B) ∈ Mr,s are disjoint. More precisely, it is shown that Φ preserves disjointness if and only if Φ is of the form

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“…Furthermore, a contractive linear operator between two C * -algebras preserves absolutely compatible elements (i.e., a△b in B A ⇒ T (a)△T (b)) if, and only if, T is a triple homomorphism. Having in mind the extensive literature on bounded linear operators between C * -algebras preserving (domain and/or range) orthogonality (cf., for example, [11,12,1,8,9]), the results in [2] inaugurate a new line to explore in the framework of preservers.…”

Section: Introductionmentioning

confidence: 93%

Absolutely compatible pairs in a von Neumann algebra

Jana

Karn

Peralta

2019

Electronic Journal of Linear Algebra

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Let a, b be elements in a unital C * -algebra with 0 ≤ a, b ≤ 1. The element a is absolutely compatible with b if |a − b| + |1 − a − b| = 1.In this note we find some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra. These characterizations are applied to measure how close is a pair of absolute compatible positive elements in the closed unit ball from being orthogonal or commutative. In the case of 2 by 2 matrices the results offer a geometric interpretation in terms of an ellipsoid determined by one of the points. The conclusions for 2 by 2 matrices are also applied to describe absolutely compatible pairs of positive elements in the closed unit ball of Mn.1 2 for the absolute value of a. Several attempts to establish a non-commutative version of the celebrated Kakutani's theorem [3], which characterizes those Banach lattices which are lattice isomorphic to the space C(Ω), of all continuous functions on a compact Hausdorff space Ω, have been pursued in recent years (cf. [4, 5, 6]). As in many previous forerunners, like the representation theory, published by Stone in [10], which characterizes C(Ω) in terms of order and its ring properties, a non-commutative Kakutani's theorem will necessarily rely on the notions of orthogonality, absolute value and order. Some discoveries have been found within this non-commutative program, for example, it is shown in [7, Proposition 4.9] that if a is an arbitrary positive element in the closed unit ball, B A , of a unital C * -algebra A and p is a projection in A, then |p − a| + |1 − p − a| = 1 if and only if a and p commute. Furthermore, two positive elements a and b in B A are orthogonal if, and only if, a + b ≤ 1 and |a − b| + |1 − a − b| = 1. The second condition gives rise to a strictly weaker notion than the usual orthogonality. Accordingly to the notation in 1991 Mathematics Subject Classification. Primary 46L10; Secondary 46B40 46L05.

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Linear disjointness preservers of operator algebras and related structures

Cited by 21 publications

References 70 publications

Linear Maps Which are Anti-derivable at Zero

Linear Maps Which are Anti-derivable at Zero

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Absolutely compatible pairs in a von Neumann algebra

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