A separable postliminal 𝐶*-algebra without maximal closed ideals

Leptin, H.

doi:10.1090/S0002-9947-1971-0281016-1

A separable postliminal $C^{\ast }$-algebra without maximal closed ideals
HTML articles powered by AMS MathViewer

by H. Leptin PDF

Trans. Amer. Math. Soc. 159 (1971), 489-496 Request permission

Abstract:

Let $G$ be the free abelian group with a countable number of generators. We construct a separable locally compact $G$-transformation space $X$ without closed minimal invariant subsets, such that the corresponding ${C^ \ast }$-algebra ${C^ \ast }(G,X)$ has the properties mentioned in the title. Using $X$ we also give an example of a transformation space $(G,Z)$ without closed minimal invariant subset, on which $G$ acts freely.

References

Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
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 Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911. MR 146297, DOI 10.2140/pjm.1962.12.885
Alain Guichardet, Caractères des algèbres de Banach involutives, Ann. Inst. Fourier (Grenoble) 13 (1963), 1–81 (French). MR 147925, DOI 10.5802/aif.130
A. Guichardet, Utilisation des sous-groupes distingués ouverts dans l’étude des représentations unitaires des groupes localement compacts, Compositio Math. 17 (1965), 1–35 (1965) (French). MR 188348
H. Leptin, Darstellungen verallgemeinerter $L^{1}$-Algebren, Invent. Math. 5 (1968), 192–215 (German). MR 236727, DOI 10.1007/BF01425550

Darstellungen verallgemeinerter ${L^1}$-Algebren

Masamichi Takesaki, Covariant representations of $C^{\ast }$-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303. MR 225179, DOI 10.1007/BF02392085
G. Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239 (French). MR 241994