Local automorphisms and derivations on certain 𝐶*-algebras

Abstract

Abstract. It is shown that continuous 2-local derivations on AF C * -algebras are derivations and surjective 2-local *-automorphisms on prime C * -algebras or on C * -algebras such that the identity element is properly infinite are *-automorphisms.A mapping φ of an algebra A into itself is called a local derivation (respectively, local automorphism) if for every A ∈ A there exists a derivation (respectively, automorphism) 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 Isometries on Spaces of Lipschitz Functions

“…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].…”

On 2-local *-automorphisms and 2-local isometries of B(H)

“…Here and in what follows, B(H) denotes the C * -algebra of all bounded linear operators on H. This remarkable result attracted serious attention and motivated a number of further investigations. We refer only to some of the related papers [1,2,3,5,7,9,10,13,14,15,16,20,21,22,23,26] and Chapter 3 in the book [24] which treats these kinds of problems.…”

A survey on local and 2-local derivations on C*- and von Neuman algebras

“…The reader is referred to [8, Lemma 2.12] for the proof of the next lemma. We culminate this subsection with a result on 2-local derivations for a certain subclass of C * -algebras (see [51]).…”

“…An equivalent definition is to say that A is an AF C * -algebra if it has an ascending sequence of finite-dimensional C * -subalgebras whose closed union is A. Theorem 6.11. [51]. Let A be an AF C * -algebra and let φ : A → A be a continuous 2-local derivation.…”