2-locl derivations on some C*-algebras

TL;DR

This paper investigates 2-local derivations on certain C*-algebras, establishing conditions under which these derivations are inner, especially when involving trace-open projections and finite von Neumann algebras.

Contribution

It introduces trace-open projections in the second dual of a C*-algebra and proves that 2-local derivations are inner under these conditions, extending to approximately 2-local derivations.

Findings

2-local derivations are inner if 1 is a T-open projection with a faithful normal semi-finite trace.

The theorem extends to approximately 2-local derivations in finite von Neumann algebras.

Conditions involving trace-open projections ensure derivation innerness.

Abstract

In this paper, we introduce the concept of trace-open projections in the second dual A** of a C*-algebra A, and we show that if there is a faithful normal semi-finite trace T on A**, and 1 is a T-open projection, then each 2-local derivation D from A to A** is an inner derivartion. We also show that this theorem hold for approximately 2-local derivations when A** is finite as a von neumann algebra.