Invertibility preserving linear maps on ℒ(𝒳)

Abstract. We show that the existence of a surjective isometry (which is merely a distance preserving map) between the unitary groups of unital C * -algebras implies the existence of a Jordan *-isomorphism between the algebras. In the case of von Neumann algebras we describe the structure of those isometries showing that any of them is extendible to a real linear Jordan *-isomorphism between the underlying algebras multiplied by a fixed unitary element. We present a result of similar spirit for the surjective Thompson isometries between the spaces of all invertible positive elements in unital C * -algebras. IntroductionThe study of linear isometries between function spaces or operator algebras has a long history dating back to the early 1930's. For an excellent comprehensive treatment of related results we refer to the two volume set [5,6]. The most fundamental and classical results of that research area are the Banach-Stone theorem describing the structure of all linear surjective isometries between the Banach spaces of continuous functions on compact Hausdorff spaces and its noncommutative generalization, Kadison's theorem [13], which describes the structure of all linear surjective isometries between general unital C * -algebras. One immediate consequence of those results is that if two C * -algebras are isometrically isomorphic as Banach spaces, then they are isometrically isomorphic as Jordan *-algebras, too. This provides a good example of how nicely the different sides (in the present case the linear algebraic -geometrical structure and the full algebraic, more precisely, Jordan *-algebraic structure) of one complex mathematical object may be connected to or interact with each other. We mention another famous result of similar spirit which also concerns isometries. This is the celebrated Mazur-Ulam theorem stating that any surjective isometry between normed real linear spaces is automatically an affine transformation (hence equals a real linear surjective isometry followed by a translation). This means that if two normed real linear spaces are isometric as metric spaces, then they are isometrically isomorphic as normed linear spaces, too.2010 Mathematics Subject Classification. 47B49, 46J10.

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“…If A, B are unital algebras and J : A → B is a surjective Jordan homomorphism, then by the proof of Proposition 1.3 in [23] we have…”

Section: Norm Isometries Of Unitary Groupsmentioning

confidence: 97%

Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C∗-algebras

Hatori

Molnár

2014

Journal of Mathematical Analysis and Applications

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“…In particular, we shall thereby obtain a brief proof of Sourour's result [10]. It should be mentioned, however, that several ideas from both [4] and [10] will be used in our proof.…”

Section: Introductionmentioning

confidence: 87%

“…It turns out that, under rather mild assumptions, Jordan homomorphisms are unital invertibility preserving maps (see e.g. [10,Proposition 1.3]). Motivated by various relevant results (such as the Gleason-Kahane-Żelazko theorem) Kaplansky [9] asked when the converse is true, that is, under which assumptions a unital invertibility preserving map must be a Jordan homomorphism.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by various relevant results (such as the Gleason-Kahane-Żelazko theorem) Kaplansky [9] asked when the converse is true, that is, under which assumptions a unital invertibility preserving map must be a Jordan homomorphism. There has been a lot of activity concerning this question; we refer the reader to some rather recent papers ( [3], [4], [6], [10]) for historical accounts. We shall now only briefly discuss those results that are closely connected with the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…Aupetit and du Mouton [4] extended this result to semisimple Banach algebras whose socle is an essential ideal (actually, they considered a slightly more general problem on maps preserving the full spectrum of each element). Finally, Sourour [10] showed that the result from [8] is true for maps that preserve invertibility (in only one direction).…”

Section: Introductionmentioning

confidence: 99%

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