Superderivations ofC *-algebras implemented by symmetric operators

The paper studies unbounded symmetric and dissipative implementations (S,G) of^*-superderivations ofC ^*-algebras . It associates with them representations _S of the domainsD() of on the deficiency spacesN(S) of the symmetric operatorsS. A link is obtained between the deficiency indicesn _±(S) ofS and the dimensions of irreducible representations of . For the case when (S,G) is a maximal implementation and max(n _±(S))<, some conditions are given for the representation _S to be semisimple and to extend to a bounded representation of .