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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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