On Maps Preserving Products

Abstract

Abstract. Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

A Note on 2-Local Maps

“…This result is connected with the well-known Hua theorem [14] and it generalizes some results of [5,10].…”

“…Now we can describe 2-local automorphisms of matrix algebras over finite-dimensional division rings using some ideas from [5,24].…”

“…It also generalizes a theorem by Chebotar et al . [5,Theorem 5.3], where 2-local automorphisms of finite-dimensional division rings K with characteristic 0 were described. It is interesting to note that the case of anti-automorphism (if n = 1) is really possible (see [5,Example 5.4]).…”

2-Local Isometries on Spaces of Lipschitz Functions

“…Proof Let g ∈ Lip(X). Since Φ is a 2-local isometry of Lip(X), there exists a surjective linear isometry Φˆ1 ,g of Lip(X) such that Φ(1) = Φˆ1 ,g (1) and Φ(g) = Φˆ1 ,g (g). Because the isometry group of Lip(X) is canonical, we have…”

“…Molnár [10] showed that every 2-local isometry of B(H) is a surjective linear isometry. Numerous papers on 2-locality have since appeared [9,11,17], and more recently [1,[5][6][7][8]18].…”