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By Ralf Meyer
Periodic cyclic homology is a homology conception for non-commutative algebras that performs the same function in non-commutative geometry as de Rham cohomology for gentle manifolds. whereas it produces sturdy effects for algebras of delicate or polynomial capabilities, it fails for higher algebras akin to such a lot Banach algebras or C*-algebras. Analytic and native cyclic homology are versions of periodic cyclic homology that paintings greater for such algebras. during this e-book, the writer develops and compares those theories, emphasizing their homological houses. This comprises the excision theorem, invariance below passage to definite 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 practical research in bornological vector areas, that's provided within the first chapters. The focal issues listed below are the connection with inductive platforms and the sensible calculus in non-commutative bornological algebras. a few chapters are extra ordinary and self sufficient of the remainder of the publication and may be of curiosity to researchers and scholars engaged on practical research and its purposes.
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Extra resources for Local and analytic cyclic homology
Let V and W be Fréchet spaces. The canonical continuous bilinear y W induces a bornological isomorphism map \ W V W ! 89. Let V and W be two Fréchet spaces. N/. §2, no. 1, p. §2, no. 1, p. 57]. 87. W / uniquely. W /; X /. We want to show that f is of the form fQ ı \ y W / ! X. 89. x/ D n2N n2N n2N Hence fQ is unique if it exists. x/ is independent of P y W the infinite series representing x. yn ; zn / D 0. n/yn ˝ zn is a null-sequence in y W . zn0 / in V and W and a null-sequence . 1 n2N y W ! X .
46. A bornology is called relatively compact or precompact if all bounded subsets are relatively compact or precompact, respectively. 45. Precompact bornologies are also studied by Henri Hogbe-Nlend and Vincenzo B. Moscatelli in , where they are called Schwartz bornologies. A relatively compact bornology is necessarily complete: if a subset is compact, then its disked hull is again compact and hence complete. For complete bornological vector spaces, there is no difference between precompact and relatively compact subsets.
Let V and W be complete locally convex topological vector spaces. W / ! W / ! W / ! W / ! V ˝ are bounded. The issue is when these maps are bornological isomorphisms. We can only expect positive results for special topological vector spaces like Fréchet spaces. In this case, we can use results of Alexander Grothendieck (). 86. Let V and W be Banach spaces. We recall the definition of V ˝ in this case and observe that ˆ0V;W is an isomorphism. Let B Â V and D Â W be the closed unit balls. B ˝ D/} .