Abstract
Given a discrete abelian group G and σ∈Ğ, the C*-crossed product\(G x_{\tilde \sigma } C (T)\) for the G-action\(\tilde \sigma _S (f) = f(< \overline {s,\sigma } > \cdot )\) on C(T) generalizes rotation C*-algebras studied by Høegh-Krohn, Skjelbred [9], Pimsner,Voiculescu [16], Riedel [18], Rieffel [19] and others. We treat\(G x_{\tilde \sigma } C (T)\) in the frame of a larger class of C*-algebras, each defined by generators with a σ-twisted commutation property. A description of such algebras as twisted C*-crossed products leads to centre, ideal lattice, primitive ideal space, tracial functionals, a characterization of simplicity and a classification under liminarity conditions. The subclass corresponding to faithful characters is classified up to isomorphism.
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References
Albeverio, S., Høegh-Krohn, R.: Ergodic actions by compact groups on C*-algebras. Math. Z.174, 1–17 (1980)
Bunce, J.: Characterizations of amenable and strongly amenable C*-algebras. Pac. J. Math.43, 563–572 (1972)
Connes, A.: On the cohomology of operator algebras. J. Funct. Anal.28, 248–253 (1978)
De Brabanter, M.: Decomposition theorems for certain C*-crossed products. Math. Proc. Camb. Philos. Soc., to appear
De Brabanter, M.: The classification of rational rotation C*-algebras. Preprint Leuven (1983)
Ghatage, P.G.: C*-algebras generated by weighted shifts. Indiana Univ. Math. J.28, 1007–1012 (1979)
Ghatage, P.G., Phillips, W.J.: C*-algebras generated by weighted shifts II. Indiana Univ. Math. J.30, 539–546 (1981)
Henrard, G.: Duality and a fixed point theorem for almost periodic C*-crossed products. Preprint Leuven (1983)
Høegh-Krohn, R., Skjelbred, T.: Classification of C*-algebras admitting ergodic actions of the two-dimensional torus. J. Reine Angew. Math.328, 1–8 (1981)
Johnson, B.E.: Cohomology in Banach algebras. Mem.Amer. Math. Soc.127; AMS, Providence R.I. (1972)
Kishimoto, A., Takai, H.: Some remarks on C*-dynamical systems with a compact abelian group. Publ. Res. Inst. Mat. Sci.14, 383–397 (1978)
Landstad, M.B.: Duality theory for covariant systems. Trans. Amer. Math. Soc.248, 223–267 (1979)
Olesen, D., Pedersen, G.K., Takesaki, M.: Ergodic actions of compact abelian groups. J. Oper. Theory3, 237–269 (1980)
Pedersen, G.K.: The linear span of projections in simple C*-algebras. J. Oper. Theory4, 289–296 (1980)
Pedersen, G.K.: C*-algebras and their automorphism groups. London Math. Soc. Monographs 14, London-New York: Academic Press 1979
Pimsner, M., Voiculescu, D.: Imbedding the irrational rotation C*-algebra into an AF-algebra. J. Oper. Theory4, 199–208 (1980)
Pontryagin, L.S.: Topological groups. 2nd Ed. London-New York: Gordon and Breach 1966
Riedel, N.: Classification of the C*-algebras associated with minimal rotations. Pac. J. Math.101, 153–161 (1982)
Rieffel, M.A.: C*-algebras associated with irrational rotations. Pac. J.Math.93, 415–429 (1981)
Rosenberg, J.: Amenability of crossed products of C*-algebras. Commun. Math. Phys.57, 187–191 (1977)
Sutherland, C.E.: Cohomology and extensions of von Neumann algebras, II. Publ. Res. Inst. Mat. Sci.16, 135–174 (1980)
Zeller-Meier, G.: Produits croisés d'une C*-algèbre par un groupe d'automorphismes. J. Math. Pures Appl. IX Sér.47, 101–239 (1968)
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De Brabanter, M., Zettl, H.H. C*-algebras associated with rotation groups and characters. Manuscripta Math 47, 153–174 (1984). https://doi.org/10.1007/BF01174591
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DOI: https://doi.org/10.1007/BF01174591