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Compact metric spaces, Fredholm modules, and hyperfiniteness. (English) Zbl 0718.46051
An unbounded p-summable Fredholm module over a \(C^*\)-algebra A is a Hilbert space H which is a left module over A (a representation of A) and a selfadjoint operator D on H such that the commutator [D,a] is a bounded operator for each \(a\in A\) and such that \((1+D^ 2)^{-p/2}\) is trace class.
The author shows that the existence of a finitely summable unbounded Fredholm module on a \(C^*\)-algebra is a relatively restrictive property. In particular, he shows that such a \(C^*\)-algebra must have a trace state and also that no such module can exist for the reduced \(C^*\)-algebra of a non-amenable discrete group.
Somewhat independently, he also shows that an unbounded Fredholm module (H,D), where \((1+D^ 2)^{-1}\) is compact, gives rise to a metric on the state space of A which generalizes the Riemannian metric in the commutative case, if one takes the canonical Fredholm module given by the Dirac operator.
Finally, he introduces the notion of a \(\Theta\)-summable unbounded Fredholm module (H,D) by the condition that \(e^{-tD^ 2}\) is trace class for some \(t>0\) and he shows that the natural word length module is \(\Theta\)-summable for each finitely generated discrete group.
In subsequent work, the author has developed the theory of entire cyclic cohomology, which allows to associate cyclic cocycles with \(\Theta\)- summable Fredholm modules.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
58B15 Fredholm structures on infinite-dimensional manifolds
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References:
[1] DOI: 10.1090/S0002-9904-1969-12235-4 · Zbl 0229.22026 · doi:10.1090/S0002-9904-1969-12235-4
[2] DOI: 10.1070/IM1974v008n01ABEH002097 · Zbl 0299.57010 · doi:10.1070/IM1974v008n01ABEH002097
[3] Connes, Contemp. Math. Operator Algebras and Math. Physics none pp 237– (1986)
[4] DOI: 10.2307/1971057 · Zbl 0343.46042 · doi:10.2307/1971057
[5] Connes, Publ. IHES 62 pp 41– (1985) · Zbl 0592.46056 · doi:10.1007/BF02698807
[6] Connes, C.R. Acad. Sci. 290 pp 599– (1980)
[7] Connes, Ergod. Th. & Dynam. Sys. 1 pp 431– (1981)
[8] DOI: 10.1016/0022-1236(77)90062-3 · Zbl 0369.28013 · doi:10.1016/0022-1236(77)90062-3
[9] Bedos, Moyennabilité intérieure des groupes, définitions et exemples
[10] Baum, Geometric (1982)
[11] Baaj, C.R. Acad. Sci. 296 pp 875– (1983)
[12] Witten, J. Diff. Geom. 17 pp 661– (1982)
[13] Kasparov, Izv. Akad. Nauk. SSSR 44 pp 571– (1980)
[14] Jaffe, Index of a family of Dirac operators on loop space (1987) · Zbl 0629.58040
[15] Hörmander, Elliptische Differentialgleischungen Band II (1969)
[16] Gromov, Publ. IHES 53 pp 53– (1981) · Zbl 0474.20018 · doi:10.1007/BF02698687
[17] Gilkey, Math. Lecture Series II pp none– (1984)
[18] DOI: 10.2307/2373108 · Zbl 0191.42803 · doi:10.2307/2373108
[19] DOI: 10.2307/2372852 · Zbl 0087.11501 · doi:10.2307/2372852
[20] Dixmier, Les Algèbres d’opérateurs dans l’espace Hilbertien 2ème édition (1969)
[21] Delorme, Bull. Soc. Math. 105 pp 281– (1977)
[22] Connes, C.R. Acad. Sc. 303 pp 913– (1986)
[23] DOI: 10.2307/2039555 · Zbl 0275.22012 · doi:10.2307/2039555
[24] Teleman, Publ. Math. IHES 58 pp 39– (1983) · Zbl 0531.58044 · doi:10.1007/BF02953772
[25] Sakai, Ergebnisse der Mathematik un ihrer Grenzgebiete 60 pp none– (none)
[26] Quillen, Topology 24 pp 89– (1985) · Zbl 0569.58030 · doi:10.1016/0040-9383(85)90047-3
[27] DOI: 10.1007/BF01645492 · Zbl 0186.28301 · doi:10.1007/BF01645492
[28] DOI: 10.1090/S0273-0979-1980-14702-3 · Zbl 0427.28018 · doi:10.1090/S0273-0979-1980-14702-3
[29] Kazhdan, Funkt. Anal. i. Priloj. 1 pp 71– (1967)
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