2-Local derivations on AW$^*$-algebras of type I

TL;DR

The paper proves that all 2-local derivations on type I AW$^*$-algebras are derivations and establishes a Gleason-type theorem for signed measures on projections in these algebras, excluding some specific types.

Contribution

It demonstrates that 2-local derivations on type I AW$^*$-algebras are actual derivations and extends Gleason's theorem to signed measures on projections in these algebras.

Findings

Every 2-local derivation on a type I AW$^*$-algebra is a derivation.

An analog of Gleason's theorem is established for signed measures on projections.

Exceptions include certain type I$_2$ and type I$_m$ ($2<m<inite$) AW$^*$-algebras.

Abstract

It is proved that every 2-local derivation on an AW$^*$-algebra of type I is a derivation. Also an analog of Gleason theorem for signed measures on projections of homogenous AW$^*$-algebras except the cases of an AW$^*$-algebra of type I$_2$ and a factor of type I$_m$, $2<m<\infty$ is proved.