Representations of $C^*$-algebras in spaces with indefinite metric

The problem is considered of orthogonalization of $J$-symmetric representations of $C^*$-algebras in the Pontryagin spaces $\Pi_\varkappa$. It is proved that in spaces with finite rank of indefiniteness, every such representation is similar to a $*$-representation in a Hilbert space. Necessary and sufficient conditions are established for the existence of an invariant dual pair of subspaces for a $J$-symmetric operator algebra.