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

Abstract

Abstract. We survey the results on local and 2-local derivations on C * -algebras, von Neumann algebras and JB * -triples.

Weak-2-local derivations on Mn

“…As in previous studies on 2-local derivations and * -homomorphisms (cf. [1,13,5,6] and [2]), the techniques in this paper rely on the Bunce-Wright-Mackey-Gleason theorem [4], however, certain subtle circumstances and pathologies, which are intrinsical to the lattice P(M n ) of all projections in M n , increase the difficulties with respect to previous contributions. More concretely, the just mentioned Bunce-Wright-Mackey-Gleason theorem asserts that every bounded, finitely additive (vector) measure on the set of projections of a von Neumann algebra M with no direct summand of Type I 2 extends (uniquely) to a bounded linear operator defined on M .…”

Boundedness of completely additive measures with application to 2-local triple derivations

“…Applications can be found in quantum physics and quantum information (cf. [18], [43], [37], [36], [22,Chapter 7], and [16], among many others), and in functional analysis with studies on vector-valued measures on von Neumann algebras and 2-local maps on von Neumann algebras, JBW * -algebras and JBW * -triples (see [20], [3], [4], [13] [14] and [32]).…”

“…For an elaboration of the above summary, see the forthcoming survey of Ayupov, Kudaybergenov, and Peralta, [4]. Local and 2-local derivations have also been considered on algebras of measurable operators associated with von Neumann algebras.…”

Characterization of centralizers on nest subalgebras of von Neumann algebras by local action

“…In general, there are two directions in the study of the local actions of mappings of operator algebras. One is the well-known local mappings problem, such as local derivations and local automorphisms (for example, see [10][11][12][13] and references therein). Motivated by local derivations and local isomorphisms, we call a linear mapping a local left centralizer of A, if for each A ∈ A there is a left centralizer A of A such that (A) = A (A).…”