The similarity problem for representations of 𝐶*-algebras

Bunce, John W.

doi:10.1090/S0002-9939-1981-0597652-0

The similarity problem for representations of $C^{\ast }$-algebras
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by John W. Bunce

Proc. Amer. Math. Soc. 81 (1981), 409-414

DOI: https://doi.org/10.1090/S0002-9939-1981-0597652-0

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Abstract:

Let $\pi :A \to B(H)$ be a bounded homomorphism of a ${C^*}$-algebra into the bounded operators on a Hilbert space. We prove that, if $\pi$ is cyclic, there is a $*$-representation $\theta :A \to B(H)$ and a bounded one-to-one positive operator $P$ such that $P\theta (a) = \pi (a)P$. We include applications to $\theta$-derivations and invariant operator ranges for operator algebras.

References

Derivations of non-separable

-algebras

Bruce A. Barnes, The similarity problem for representations of a $B^{\ast }$-algebra, Michigan Math. J. 22 (1975), 25–32. MR 372628
Bruce A. Barnes, When is a representation of a Banach $*$-algebra Naimark-related to a $*$-representation?, Pacific J. Math. 72 (1977), no. 1, 5–25. MR 458182

Representations Naimark-related to

-representations; a correction

John Bunce, Respresentations of strongly amenable $C^{\ast }$-algebras, Proc. Amer. Math. Soc. 32 (1972), 241–246. MR 295091, DOI 10.1090/S0002-9939-1972-0295091-8
Erik Christensen, Extension of derivations, J. Functional Analysis 27 (1978), no. 2, 234–247. MR 481217, DOI 10.1016/0022-1236(78)90029-0

Extensions of derivations

Jacques Dixmier, Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. (Szeged) 12 (1950), 213–227 (French). MR 37470
George A. Elliott, On approximately finite-dimensional von Neumann algebras. II, Canad. Math. Bull. 21 (1978), no. 4, 415–418. MR 523582, DOI 10.4153/CMB-1978-073-1
Ciprian Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 21 (1971/72), 887–906. MR 293439, DOI 10.1512/iumj.1972.21.21072
Richard V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 77 (1955), 600–620. MR 72442, DOI 10.2307/2372645
Gilles Pisier, Grothendieck’s theorem for noncommutative $C^{\ast }$-algebras, with an appendix on Grothendieck’s constants, J. Functional Analysis 29 (1978), no. 3, 397–415. MR 512252, DOI 10.1016/0022-1236(78)90038-1
J. R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972), 432–438. MR 374927, DOI 10.1112/jlms/s2-5.3.432
J. R. Ringrose, Linear mappings between operator algebras, Symposia Mathematica, Vol. XX (Convegno sulle Algebre $C^*$ e loro Applicazioni in Fisica Teorica, Convegno sulla Teoria degli Operatori Indice e Teoria $K$, INDAM, Rome, 1975) Academic Press, London, 1976, pp. 297–315. MR 0461159