A survey on Tingley’s problem for operator algebras

Abstract

We survey the most recent results on extension of isometries between special subsets of the unit spheres of C * -algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras, and p-Schattenvon Neumann spaces, with special interest on Tingley's problem.Problem 1.1. Let X and Y be two Banach spaces whose unit spheres are denoted by S(X) and S(Y ), respectively. Let S 1 and S 2 be two subsets of S(X) and S(Y ), respectively. Suppose ∆ : S 1 → S 2 is a surjective isometry. Does ∆ extend to a real linear isometry from X onto Y ?Henceforth, we shall write T for the unit sphere of C. The complex conjugation on T cannot be extended to a complex linear isometry on C. So, in the case of complex Banach spaces, a complex linear extension is simply hopeless for all cases. Similar constrains will appear in subsequent results.These problems, whose origins are in geometry, are nowadays a central topic for those researchers working on preservers. If in Problem 1.1 we consider S 1 = S(X) and S 2

Tingley's problem for $p$-Schatten von Neumann classes

“…The most recent achievement in this line is a result by M. Mori, which establishes that a surjective isometry between the unit spheres of two von Neumann algebra preduals admits a unique extension to a surjective real linear isometry between the corresponding spaces [25]. We refer to the surveys [9,28,40] for a detailed overview on Tingley's problem.…”

A reflection on Tingley's problem and some applications

“…This problem has been dealt in several ways and lots of positive answers have been found, see, e.g., [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] -it is really impressive the development of machinery and technics that this problem has led to.…”

On the unit sphere of positive operators

“…During the last thirty years, mathematicians have pursued an argument to prove or discard a positive solution to Tingley's problem (compare the survey [26]). This problem, in which Geometry and Functional Analysis interplay, is just as attractive as difficult.…”

“…The most recent achievements in this line establish that a surjective isometry between the unit spheres of two arbitrary von Neumann algebras admits a unique extension to a surjective real linear isometry between the corresponding von Neumann algebras [14], and an excellent contribution due to M. Mori contains a complete positive solution to Tingley's problem for surjective isometries between the unit spheres of von Neumann algebra preduals [21]. Readers interested in learning more details can consult the recent survey [26].…”