# Local and Analytic Cyclic Homology (EMS Tracts in - download pdf or read online

By Ralf Meyer

Periodic cyclic homology is a homology concept for non-commutative algebras that performs an identical position in non-commutative geometry as de Rham cohomology for delicate manifolds. whereas it produces reliable effects for algebras of delicate or polynomial capabilities, it fails for greater algebras corresponding to so much Banach algebras or C*-algebras. Analytic and native cyclic homology are versions of periodic cyclic homology that paintings greater for such algebras. during this ebook, the writer develops and compares those theories, emphasizing their homological homes. This comprises the excision theorem, invariance less than passage to convinced dense subalgebras, a common Coefficient Theorem that relates them to $K$-theory, and the Chern-Connes personality for $K$-theory and $K$-homology. The cyclic homology theories studied during this textual content require a great deal of sensible research in bornological vector areas, that's provided within the first chapters. The focal issues listed below are the connection with inductive structures and the useful calculus in non-commutative bornological algebras. a few chapters are extra uncomplicated and autonomous of the remainder of the publication and may be of curiosity to researchers and scholars engaged on practical research and its functions.

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**Example text**

W be a bounded linear map with complete range W . We claim that l D 0. Let V" be the completion of V with respect to the norm " . 0; "/. Since W is complete, we can extend l to a bounded linear map l" W V" ! W . Pick a continuous function f 2 V" vanishing in a neighbourhood of 0. Then f is annihilated by the restriction map r";"0 W V" ! 0; ". However, l" D l"0 ı r";"0 factors through this restriction map. f / D 0. 0/ D 0 for all f 2 V" , the space of functions vanishing in a neighbourhood of 0 is dense in V" .

A; B/ for the set of algebra homomorphisms A ! C/. 106. An algebra is unital if it comes equipped with a map u W 1 ! A called unit such that the diagram idA ˝u u˝idA GA˝Ao A ˝ 1P 1˝A PPP nn PPŠP Š nnn m nnn can PPPP PP9 wnnnnn can A commutes; the isomorphisms A ˝ 1 Š A Š 1 ˝ A come from the symmetric monoidal category structure. An algebra homomorphism f W A ! B between two unital algebras is unital if u f 1 ! A ! B agrees with the unit for B. A; B/ for the set of unital algebra homomorphisms A !

V /. M; V / for the space of C k -functions M ! V . We will only use continuous functions with values in complete bornological vector spaces. X; V /. These bornologies are complete or separated if V is so. X / is a Banach space; the uniformly bounded bornology is its von Neumann bornology, the uniformly continuous bornology is its precompact bornology by the Arzelà–Ascoli Theorem. X; V / with the uniformly continuous bornology. g with a canonical bornology. S / is a uniformly continuous set of functions M !