General Mazur–Ulam Type Theorems and Some Applications

Abstract

ABSTRACT. Recently we have presented several structural results on certain isometries of spaces of positive definite matrices and on those of unitary groups. The aim of this paper is to put those previous results into a common perspective and extend them to the context of operator algebras, namely, to that of von Neumann factors.

Transformations Preserving Norms of Means of Positive Operators and Nonnegative Functions

“…It is clear that 0 = v = xy ∈ A + and v ≤ x, y. Therefore, we have the equivalence in the above displayed formula and then (18) follows. Next we show that for any peaking function x ∈ A + which peaks at a point t ∈ X, the function φ(x) is also a peaking function that peaks at some point ϕ(t) ∈ Y .…”

“…Indeed, it follows from the monotonicity of p-norms which is a wellknown fact. (For a statement of similar spirit concerning general symmetric norms on C * -algebras that we shall use later, see Lemma 12 in [18].) To verify the sufficiency part we first assert that…”

“…By Lemma 12 in [18] we know that any symmetric norm on any C * -algebra is monotone on the positive cone. This together with the monotonicity property (O1) of operator means imply that if A, B in A + and A ≤ B then it follows that N (AσC) ≤ N (BσC) holds for every C ∈ A + .…”

“…On the other hand, by (18), φ preserves zero product in both directions which implies that φ −1 (u)φ −1 (v) = 0. Therefore, the functions φ −1 (u) and φ −1 (v) of norm 1 take the value 1 at two different points and then, by (19), the same must be true for x.…”

Continuous Jordan triple endomorphisms of P2

“…Our main reason for investigating those maps comes from the fact that they naturally appear in the study of surjective isometries and surjective maps preserving generalized distance measures between positive definite cones. For details see [9,10,11].In the paper [9] we have proved the following statement which appeared as Theorem 1 there. In what follows we denote by M n the algebra of all n × n complex matrices and P n stands for the cone of all positive definite matrices in M n .…”