Geodesic
Representations of $C^*$-algebras in spaces with indefinite metric
Matematičeskie zametki, Tome 22 (1977) no. 4, pp. 583-592.

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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. 
@article{MZM_1977_22_4_a14,
     author = {V. S. Shulman},
     title = {Representations of $C^*$-algebras in spaces with indefinite metric},
     journal = {Matemati\v{c}eskie zametki},
     pages = {583--592},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_4_a14/}
}
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V. S. Shulman. Representations of $C^*$-algebras in spaces with indefinite metric. Matematičeskie zametki, Tome 22 (1977) no. 4, pp. 583-592. http://geodesic.mathdoc.fr/item/MZM_1977_22_4_a14/