Isometries and Maps Compatible with Inverted Jordan Triple Products on Groups

Hatori, Osamu; Hirasawa, Go; Miura, Takeshi; Molnár, Lajos

doi:10.3836/tjm/1358951327

Cited by 23 publications

(41 citation statements)

References 12 publications

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“…This is what we have called in our recent paper [13] inverted Jordan triple product and defined as (A, B) → AB −1 A. The reason to introduce that operation is the following.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The reason to introduce that operation is the following. In [13] we have made attempts to generalize Mazur-Ulam theorem in a non-commutative context. That famous result states that the surjective isometries (i.e., surjective distance preserving maps) between normed real linear spaces are automatically affine.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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Jordan triple endomorphisms and isometries of spaces of positive definite matrices

Molnár

2013

Linear and Multilinear Algebra

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Abstract. In this paper we determine the structure of certain algebraic morphisms and isometries of the space Pn of all n × n complex positive definite matrices. In the case n ≥ 3 we describe all continuous Jordan triple endomorphisms of Pn which are continuous maps φ : Pn → Pn satisfyingIt has recently been discovered that surjective isometries of certain substructures of groups equipped with metrics which are in a way compatible with the group operations have algebraic properties that relate them rather closely to Jordan triple morphisms. This makes us possible to use our structural results to describe all surjective isometries of Pn that correspond to any member of a large class of metrics generalizing the geodesic distance in the natural Riemannian structure on Pn. Finally, we determine the isometry group of Pn relative to a very recently introduced metric that originates from the divergence called Stein's loss.

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“…This is what we have called in our recent paper [13] inverted Jordan triple product and defined as (A, B) → AB −1 A. The reason to introduce that operation is the following.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Jordan triple endomorphisms and isometries of spaces of positive definite matrices

Molnár

2013

Linear and Multilinear Algebra

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“…The most important one among them, Proposition 11, shows that on certain substructures of groups surjective transformations that preserve a given generalized distance measure d which is compatible with the group operation, necessarily preserve locally the so-called inverted Jordan triple product (i.e., they respect the operation xy −1 x). We point out that results of this kind (which can be considered as noncommutative versions of the famous MazurUlam theorem) are first appeared in the paper [12]. In fact, below we closely follow the approach presented in Sections 2 and 3 of that paper but here we have to make several small modifications according to our present need.…”

Section: Proofsmentioning

confidence: 99%

“…in Definitions 3.2 and 3.4 in that paper. Finally, we shall obtain Proposition 11, a statement similar to Corollary 3.10 in [12] which is the basic tool in the proof of our main result. holds for any x, y ∈ X.…”

Section: Proofsmentioning

confidence: 99%

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Transformations on positive definite matrices preserving generalized distance measures

Molnár

Szokol

2015

Linear Algebra and its Applications

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Abstract. We substantially extend and unify former results on the structure of surjective isometries of spaces of positive definite matrices obtained in the paper [14]. The isometries there correspond to certain geodesic distances in Finsler-type structures and to a recently defined interesting metric which also follows a nonEuclidean geometry. The novelty in our present paper is that here we consider not only true metrics but so-called generalized distance measures which are parameterized by unitarily invariant norms and continuous real functions satisfying certain conditions. Among the many possible applications, we shall see that using our new result it is easy to describe the surjective maps of the set of positive definite matrices that preserve the Stein's loss or several other types of divergences. We also present results concerning similar preserver transformations defined on the subset of all complex positive definite matrices with unit determinant.

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