Isometries on Banach Spaces

Isometries on positive operators of unit norm

“…The natural morphisms of metric spaces are the isometries, they have a vast literature. Concerning results on linear isometries of normed spaces we refer to the two volume set [2,3]. In the case when a linear structure is not present, the problem of isometries becomes much more difficult.…”

Spectrally isometric elementary operators

“…While it is difficult to control the behaviour of T 1 when T is just a spectrally bounded operator (see the discussion in [15]), if T is a surjective spectral isometry, then T 1 is central and σ(T 1) is always contained in the unit circle T ([19, Proposition 2.3]). By the afore-mentioned method, this follows immediately from a description of non-unital isometries on subalgebras of algebras of continuous functions due to deLeeuw-Rudin-Wermer ( [11,Corollary 2.3.16]). Replacing T by x → (T 1) −1 T x, x ∈ A, if necessary, we can henceforth assume that our spectral isometries are unital.…”

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

“…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.…”