Extension of derivations in the algebra of compact operators

Abstract

Let C( H) denote the C ∗-algebra of all compact linear operators on a complex Hilbert space H. If δ is a closable ∗-derivation in C( H) which anti-commutes with an involutive ∗-antiautomorphism α and has finite spatial deficiency-indices, then there exists an infinitesimal generator δ 0 of a continuous action of R on C( H) which extends δ and anti-commutes with α. This is an analogue of the von Neumann's theorem which states that a symmetric operator commuting with a conjugation J has a self-adjoint extension which also commutes with J.