Ralf Meyer Analytic cyclic cohomology 1999 arXiv:math/9906205v1 [math.KT] 29 Jun 1999 Contents 1 Introduction 4 2 Bornologies 8 2.1 BasicDefinitions ................................. .... 9 2.1.1 Bornologies ................................... 9 2.1.2 Boundedmaps .................................. 10 2.1.3 Bornologicalconvergence . ..... 11 2.2 Constructions with bornological vector spaces . .............. 12 2.2.1 Subspaces,quotients,extensions . ........ 12 2.2.2 Completions ................................... 13 2.2.3 Completed bornologicaltensor products . ......... 14 2.2.4 Spacesofboundedlinearmaps . .... 16 2.2.5 Smooth and absolutely continuous homotopies . ......... 16 3 Analytic cyclic cohomology 18 3.1 Analytic tensor algebras and a-nilpotent algebras . ............... 18 3.1.1 Definition of the analytic tensor algebra . ......... 21 3.1.2 Properties of analytically nilpotent algebras . ............. 23 3.1.3 The interrelations between analytic nilpotence, lanilcurs, and analytic tensor algebras...................................... 25 3.1.4 Reformulations of the Extension and the Homotopy Axiom ......... 28 3.1.5 Universal analytically nilpotent extensions . ............. 30 3.1.6 The bimodule Ω1(T A).............................. 32 3.1.7 The lanilcur category and Goodwillie’s theorem . ........... 34 3.2 The X-complex of T A andanalyticcycliccohomology . 36 3.2.1 TheX-complexofaquasi-freealgebra . ....... 36 3.2.2 Definition and functoriality of analytic cyclic cohomology .......... 37 3.2.3 Homotopy invariance and stability . ....... 38 3.2.4 Adjoiningunits................................ .. 40 3.2.5 The Chern-Connes character in K-theory ................... 42 3.2.6 The X-complex of T A andentirecycliccohomology . 44 3.3 Excisioninanalyticcycliccohomology . ........... 46 3.3.1 Outlineoftheproof ............................. .. 47 3.3.2 Linear functoriality of Ωan ............................ 48 3.3.3 Someisomorphisms .............................. 49 3.3.4 A free resolution of L+ .............................. 51 3.3.5 Analytic quasi-freeness of L ........................... 53 3.4 The Chern-Connes character in K-homology ..................... 57 2 CONTENTS 3 3.4.1 Thefinitelysummablecase . ... 58 3.4.2 The metric approximation property and cut-off sequences .......... 62 3.4.3 Construction of the Chern-Connes character. .......... 63 3.4.4 K-homologyandtheindexpairing . 66 3.5 Tensoring algebras and the exterior product . ............ 68 4 Periodic cyclic cohomology 71 4.1 Projectivesystems ............................... ..... 73 4.2 Periodic cyclic cohomology for pro-algebras . ............. 74 4.3 Dimension estimates in the Excision Theorem . ........... 78 4.4 Fr´echetalgebrasandfinealgebras. .......... 85 A Appendices 87 A.1 Bornologiesandinductivesystems . ......... 87 A.1.1 Functorial constructions with bornological vector spaces........... 87 A.1.2 Bornological vector spaces and inductive systems . ............ 88 A.1.3 Constructionofcompletions. ...... 90 A.1.4 TensorproductsofFr´echetspaces . ....... 91 A.1.5 Admissible Fr´echet algebras . ....... 92 A.2 Homological algebra and universal algebra . ............ 93 A.2.1 Complexes of complete bornological vector spaces . ............ 94 A.2.2 Modules over a complete bornological algebra A ............... 95 A.2.3 The bar resolution and Hochschild homology . ........ 95 A.2.4 The tensor algebra and differential forms . ........ 98 A.2.5 Quasi-freealgebras. .... 99 A.3 Cyclichomologyandcohomology . ....... 100 A.3.1 TheKaroubioperator .. .. .. .. .. .. .. .. .. .. .. .. 101 A.3.2 The Hodge filtration and cyclic cohomology . ........ 102 A.4 The homotopy invariance of the X-complex . ......... 104 A.4.1 Contracting the Hochschild complex . ....... 105 A.4.2 Absolutely continuous homotopies and chain homotopies........... 106 A.5 Some computations in X(T A).............................. 107 A.5.1 The boundary of X(T A)............................. 107 A.5.2 Boundednessofthe spectraloperators . ........ 108 Chapter 1 Introduction The main results of this thesis are a proof of excision for entire cyclic cohomology and the con- struction of a Chern-Connes character for Fredholm modules without any summability conditions having values in a variant of entire cyclic cohomology. The excision theorem in entire cyclic cohomology asserts that if K ֌ E ։ Q is an algebra extension with a bounded linear section s: Q → E, then there is a six-term exact sequence ∗ ∗ i p HE0(Q) / HE0(E) / HE0(K) O 1 1 1 HE (K) o ∗ HE (E) o ∗ HE (Q) p i of the associated entire cyclic cohomology groups HE∗(xy). Recently, this has been proved also by Puschnigg [34] using quite different methods. Entire cyclic cohomology is a relative of periodic cyclic cohomology that accommodates also certain “infinite dimensional” cohomology contributions. This can be made more precise in terms of the Chern-Connes character in K-homology. In periodic cyclic cohomology, the Chern-Connes character can be defined only for finitely summable Fredholm modules. That is, the commutators [x, F ] that are required to be compact for a Fredholm module are even contained in a Schatten ideal ℓp(H) for some p. Unfortunately, there are many interesting Fredholm modules that are not finitely summable. In entire cyclic cohomology, the summability condition can be weakened to θ-summability [7]. This allows certain “infinite” Fredholm modules, but still imposes a serious restriction. In particular, Fredholm modules over C∗-algebras usually are not θ-summable. However, a small modification of the definition of entire cyclic cohomology allows to define a Chern-Connes character for Fredholm modules without any summability restriction. For entire ⊗n cyclic cohomology we consider families of multi-linear maps φn : A → C, n ∈ N, such that for all bounded sets S ⊂ A, there is a constant C(S) such that |φn(a1,..., an)| ≤ C for all n ∈ N, a1,..., an ∈ S. If we instead require this “entire growth” condition only for compact sets S, we get more linear maps. It turns out that we get sufficiently many cochains to write down a Chern- Connes character for Fredholm modules without imposing any summability condition (Section 3.4). The resulting character coincides with the usual character for finitely summable Fredholm modules and probably also for θ-summable Fredholm modules. However, I have not checked the latter. If we impose the entire growth condition only on finite subsets instead of bounded or compact subsets, we obtain a theory that is considered already by Connes [8]. 4 5 So far we have not specified for which algebras entire cyclic cohomology is defined. Usually, entire cyclic cohomology is defined on locally convex topological algebras. However, only the bounded subsets are used in its definition to formulate the growth constraint mentioned above. Moreover, we may use the compact or the finite subsets instead of the bounded subsets. Therefore, a more natural domain of definition is the category of complete bornological algebras. A bornological vector space is a vector space V together with a collection S of subsets, satisfying certain axioms already formulated by Bourbaki [4]. A standard example is the collection of all bounded subsets of a locally convex topological vector space. We call sets S ∈ S small. It appears that bornological vector spaces have not yet been used by people studying cyclic cohomology theories. Hence we outline some basic analysis in Chapter 2. We do not need much analysis, because we really want to do algebra. Analysis mainly has to set the stage for the algebra to go through smoothly. It is impossible to understand entire cyclic cohomology using only topological vector spaces. Unless A is a Banach algebra, it appears to be impossible to describe the semi-norms of a locally convex topology on A⊗ˆ n such that the continuous linear functionals are precisely the families of multi-linear mapsP satisfying the entire growth condition. In adittion, for the purposes of the Chern-Connes character it is reasonable to endow a C∗-algebra with the bornology generated by the compact subsets instead of the bornology of all bounded subsets. This change of bornology cannot be described as choosing a different topology. The only reasonable alternative to bornological vector spaces are inductive systems. These are used quite successfully by Puschnigg in [33] and are necessary to handle local theories. However, for the purposes of entire cyclic cohomology it suffices to work with bornological vector spaces. We will see in Appendix A.1.2 that inductive systems and bornological vector spaces are closely related. The original definition of periodic cyclic cohomology by Connes suggests the following route to compute the periodic cyclic cohomology of an algebra A. Firstly, compute the Hochschild cohomol- ogy HH∗(A). Secondly, use Connes’s long exact “SBI” sequence to obtain the cyclic cohomology HC∗(A) from HH∗(A). Finally, the periodic cyclic cohomology HP∗(A) is the inductive limit HP∗(A) = lim HC∗+2k(A). (1.1) −→ This powerful strategy works, for example, for smooth manifolds and group algebras. However, it fails miserably if we cannot compute the Hochschild cohomology of A. This happens, for instance, for the Schatten ideals ℓp(H) with p ≫ 1. Nevertheless, Cuntz [9] was able to compute HP0(ℓp)= C and HP1(ℓp) = 0 for all p without knowing anything about the Hochschild or cyclic cohomology of ℓp(H). This is possible because periodic cyclic cohomology has better homological properties than Hochschild or cyclic cohomology. It is invariant under smooth