Topological dynamics and 𝐶*-algebras

Green, William L.

doi:10.1090/S0002-9947-1975-0383091-6

Topological dynamics and $C^{\ast }$-algebras
HTML articles powered by AMS MathViewer

by William L. Green

Trans. Amer. Math. Soc. 210 (1975), 107-121

DOI: https://doi.org/10.1090/S0002-9947-1975-0383091-6

PDF | Request permission

Abstract:

If $G$ is a group of automorphisms of a ${C^ \ast }$-algebra $A$ with identity, then $G$ acts in a natural way as a transformation group on the state space $S(A)$ of $A$. Moreover, this action is uniformly almost periodic if and only if $G$ has compact pointwise closure in the space of all maps of $A$ into $A$. Consideration of the enveloping semigroup of $(S(A),G)$ shows that, in this case, this pointwise closure $\bar G$ is a compact topological group consisting of automorphisms of $A$. The Haar measure on $\bar G$ is used to define an analogue of the canonical center-valued trace on a finite von Neumann algebra. If $A$ possesses a sufficiently large group ${G_0}$ of inner automorphisms such that $(S(A),{G_0})$ is uniformly almost periodic, then $A$ is a central ${C^ \ast }$-algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional ${C^ \ast }$-algebra possess exactly one finite trace.

References

Johan F. Aarnes, Edward G. Effros, and Ole A. Nielsen, Locally compact spaces and two classes of $C^{\ast }$-algebras, Pacific J. Math. 34 (1970), 1–16. MR 271745, DOI 10.2140/pjm.1970.34.1
Erik M. Alfsen, Compact convex sets and boundary integrals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57, Springer-Verlag, New York-Heidelberg, 1971. MR 0445271, DOI 10.1007/978-3-642-65009-3
Ola Bratteli, Inductive limits of finite dimensional $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. MR 312282, DOI 10.1090/S0002-9947-1972-0312282-2
Jacques Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (algèbres de von Neumann), Cahiers Scientifiques, Fasc. XXV, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition, revue et augmentée. MR 0352996
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
Edward G. Effros and Frank Hahn, Locally compact transformation groups and $C^{\ast }$- algebras, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR 0227310
James G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 112057, DOI 10.1090/S0002-9947-1960-0112057-5
Siegfried Grosser and Martin Moskowitz, On central topological groups, Trans. Amer. Math. Soc. 127 (1967), 317–340. MR 209394, DOI 10.1090/S0002-9947-1967-0209394-9
Richard V. Kadison, The trace in finite operator algebras, Proc. Amer. Math. Soc. 12 (1961), 973–977. MR 139961, DOI 10.1090/S0002-9939-1961-0139961-2
Richard V. Kadison, Transformations of states in operator theory and dynamics, Topology 3 (1965), no. suppl, suppl. 2, 177–198. MR 169073, DOI 10.1016/0040-9383(65)90075-3
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Richard D. Mosak, The $L^{1}$- and $C^{\ast }$-algebras of $[FIA]^{-}_{B}$ groups, and their representations, Trans. Amer. Math. Soc. 163 (1972), 277–310. MR 293016, DOI 10.1090/S0002-9947-1972-0293016-7
B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3) 1 (1951), 178–187. MR 43779, DOI 10.1112/plms/s3-1.1.178
John C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. 58 (1952), 116–136. MR 47262, DOI 10.1090/S0002-9904-1952-09580-X
L. S. Pontriaguin, Continuous groups, Editorial Mir, Moscow, 1978 (Spanish). Translated from the third Russian edition by Carlos Vega. MR 528130
Erling Størmer, Invariant states of von Neumann algebras, Math. Scand. 30 (1972), 253–256. MR 318909, DOI 10.7146/math.scand.a-11080
Erling Størmer, Large groups of automorphisms of $C^{\ast }$-algebras, Comm. Math. Phys. 5 (1967), 1–22. MR 226422, DOI 10.1007/BF01646355
Elmar Thoma, Zur harmonischen Analyse klassenfiniter Gruppen, Invent. Math. 3 (1967), 20–42 (German). MR 213479, DOI 10.1007/BF01425489
Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824