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 delicate manifolds. whereas it produces solid effects for algebras of soft or polynomial capabilities, it fails for greater algebras akin to so much Banach algebras or C*-algebras. Analytic and native cyclic homology are variations of periodic cyclic homology that paintings higher for such algebras. during this booklet, the writer develops and compares those theories, emphasizing their homological houses. This contains 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 useful research in bornological vector areas, that is provided within the first chapters. The focal issues listed here are the connection with inductive platforms and the practical calculus in non-commutative bornological algebras. a few chapters are extra common and autonomous of the remainder of the booklet and may be of curiosity to researchers and scholars engaged on practical research and its purposes.
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V / . Similar assertions hold for C k -functions M ! g and any compact C k -manifold M . M; V / : Proof. Most of these assertions are verified in ; since the proofs are rather technical, we do not reproduce them here. 8]. The assertion about absolutely summable sequences amounts P to the well-known statement that we can rewrite an absolutely summable series n2N xn in a Fréchet space in the form xn D n xn0 with . xn0 / in V . 7]. We remark that  considers maps defined on metric spaces instead of compact spaces.
Y/ for all x 2 V , y 2 W . The maximal norm with this property is, by definition, the projective tensor product norm of . V ˝ y W /. 87. 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.
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/} . W / is a normed space equipped with its von Neumann bornology. y/ for all x 2 V , y 2 W . The maximal norm with this property is, by definition, the projective tensor product norm of . V ˝ y W /. 87. 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.