The paper studies some
properties of J-symmetric representations of ∗-algebras on indefinite metric spaces.
Making use of this, it defines the index ind(δ,S) of a ∗-derivation δ of a C∗-algebra 𝒜
relative to a symmetric implementation S of δ. The index consists of six integers
which characterize the J-symmetric representation πS of the domain D(δ) of
δ on the deficiency space N(S) of the operator S. The paper proves the
stability of the index under bounded perburbations of the derivation and that,
under certain conditions on δ, ind(δ,S) has the same value for all maximal
symmetric implementations S of δ. It applies the developed methods to the
problem of the classification of symmetric operators with deficiency indices
(1,1).
