Local automorphisms and derivations on $\mathbb {M}_n$

Abstract

Abstract. The aim of this note is to give a short proof that 2-local derivations on Mn, the n × n matrix algebra over the complex numbers are derivations and to give a shorter proof that 2-local *-automorphisms on Mn are *-automorphisms.A mapping φ of an algebra A into itself is called a local automorphism (respectively, local derivation) if for every A ∈ A there exists an automorphism (respectively, local derivation) φ A of A, depending on A, such that φ(A) = φ A (A). These notions were introduced by Kadison [Kad] and Larson and Sourour [LaSo]. In fact, their definitions were stronger. They have assumed that these mappings are also linear. Larson and Sourour proved that every local derivation on B(X), the algebra of all bounded linear operators on a Banach space X, is a derivation, and provided that X is infinite dimensional, every surjective linear local automorphism of B(X) is an automorphism. In [BrSe], they proved that the surjectivity assumption in the last result can be dropped if X is a separable Hilbert space.It is easy to see that if we drop the assumption of linearity of the local maps, then the corresponding statements are no longer true. However, in [KoSl]

2-Local Derivations on Semi-Finite Von Neumann Algebras

“…Recall that a 2-local derivation is defined as follows: Given an algebra A, a map : A → A (not linear in general) is called a 2-local derivation if for every x, y ∈ A, there exists a derivation D x,y : A → A such that (x) = D x,y (x) and (y) = D x,y (y). In 1997,Šemrl [7] introduced the notion of 2-local derivation and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinitedimensional separable Hilbert space H. A similar description for the finite-dimensional case appeared later in [5]. In the paper by Lin and Wong [6], 2-local derivations have been described on matrix algebras over finite-dimensional division rings.…”

“…In [2] the authors suggested a new technique and have generalized the abovementioned results of [7] and [5] for arbitrary Hilbert spaces, namely they considered 2-local derivations on the algebra B(H) of all linear-bounded operators on an arbitrary (no separability is assumed) Hilbert space H and proved that every 2-local derivation on B(H) is a derivation.…”

“…In [1] we also suggested another technique and generalized the above-mentioned results of [7], [5] and [2] for arbitrary von Neumann algebras of type I and proved that every 2-local derivation on these algebras is a derivation. In [3] (Theorem 3.4) a similar result was proved for finite von Neumann algebras.…”

2-Local automorphisms on finite-dimensional Lie algebras

“…In [14], P.Šemrl described 2-local derivations and 2-local automorphisms on the algebra B(H) of all bounded linear operators on the infinitedimensional separable Hilbert space H. Namely, he has proved that every 2-local automorphism (respectively, 2-local derivation) on B(H) is an automorphism (respectively, a derivation). A similar result for finite-dimensional case appeared later in [12]. Further, in [1], a new techniques was introduced to prove the same result for an arbitrary Hilbert space H (no separability is assumed).…”

Local and 2-local derivations on noncommutative Arens algebras

“…In [20], P. Semrl described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. A similar description for the finite-dimensional case appeared later in [12], [15]. In the paper [14] 2-local derivations have been described on matrix algebras over finite-dimensional division rings.…”