This book consists of the following three parts: “Cyclic theory, bivariant \(K\)-Theory and the bivariant Chern-Connes character” by J. Cuntz; “Cyclic homology” by B. Tsygan; and “Noncommutative geometry, the transverse signature operator, and Hopf algebras (after A. Connes and H. Moscovici)” by G. Skandalis (translation of [Astérisque 282, 345–364 (2002; Zbl 1035.58007)] by R. Ponge and N. Wright). They give complete accounts of cyclic theory from different and complementary points of view.
The first and the second parts both deal with cyclic theory. But the first part aimed to present the bivariant Chern–Connes character on the category of \(m\)-algebras (algebras that can be represented as projective limits of Banach algebras), and infinite-dimensional cyclic theory [J. Cuntz, Doc. Math. J. DMV 2, 139–182 (1997; Zbl 0920.19004), hereafter referred to as [1], R. Meyer, J. Eur. Math. Soc. (JEMS) 3, 269–286 (2001; Zbl 0990.19002)]. While the second part aimed to present Tamarkin–Tsygan’s noncommutative differential calculus and applications of cyclic theory to the deformation quantization, especially to the index theorem for deformation [D. Tamarkin and B. Tsygan, Lett. Math. Phys. 56, 85–97 (2001; Zbl 1008.19002); hereafter referred to as [2], R. Nest and B. Tsygan, Commun. Math. Phys. 172, 223–262 (1995; Zbl 0887.58050), hereafter referred to as [3], B. V. Fedosov, Math. Top. 5, 277-297 (1994; Zbl 0809.58012), hereafter referred to as [4]].
The third part is aimed to expose the local index theorem of A. Connes and H. Moscovici [Geom. Funct. Anal. 5, 174–243 (1995; Zbl 0960.46048), hereafter referred to as [5], Commun. Math. Phys, 48, 97–108 (1999; Zbl 0941.16024), hereafter referred to as [6]].
Besides the introduction, the first part consists of four chapters entitled “2. Cyclic Theory”, “3. Cyclic Theory for Locally Convex Algebras”, “4. Bivariant \(K\)-Theory”, “5. Infinite-dimensional Cyclic Theories”, and “Appendix A. Locally convex algebras”, “B. Standard extensions”. Chapter 2 presents cyclic theory for algebras (over \(\mathbb{C})\) without considering topology, by using three different arguments; namely, the cyclic bicomplex with various realizations, the Connes complex \(C^n_\lambda\) and the \(X\)-complex of any complete quasi-free extensions of the given algebra \(A\) [J. Cuntz and D. Quillen, J. Am. Math. Soc. 8, 251–289 (1995; Zbl 0838.19001); ibid. 373–442 (1995; Zbl 0838.19002); Invent. Math. 127, 67–98 (1997; Zbl 0889.46054)]. The author says that the \(X\)-complex yields a radically different approach to cyclic theory which is particularly useful for the study of periodic theories and cyclic theories for topological algebras, for proving the excision theorem, and for understanding the connection with topological \(K\)-theory (Section 2.5). After showing the coincidence of the periodic cyclic cohomology and de Rham cohomology as the homology of \(\mathbb{Z}/2\)-graded complexes, if \(A\) is a unital smooth algebra of homological dimension \(n\) (Theorem 2.39), homotopy invariance and Morita invariance of cyclic theories (Sections 2.7–8), and the excision theorem (Section 2.9), this chapter is concluded by computing the Chern characters of the generators \(e\) of \(HP_*\mathbb{C}\cong\mathbb{C}\) and \(u\) of \(HP_2\mathbb{C}[z, z^{-1}]\cong \mathbb{C}\) as the elements of \(HP_i(M_n(A))\cong HP_i(A)\), \(i= 0,1\) (Section 2.10). These algebraic theories are carried over directly to \(m\)-algebras in Chapter 3. Examples of \(m\)-algebras are collected in Appendix A. Necessary modifications of algebraic definitions in Chapter 2 to \(m\)-algebras are listed in Section 3.2. If \(M\) is a smooth compact manifold of dimension \(n\), then \(HP_*(C^\infty M)\cong H^{dR}_*(C^\infty M)\) and \(HC_k(C^\infty M)\cong H^{dR}_*(C^\infty M)\), \(k> n\), \(*\equiv k\text{\,mod\,}2\) (Theorem 3.2). In Section 3.3, by using the excision property of periodic cyclic homology, \(HP_*({\mathfrak l}^p(H))\cong HP_*(\mathbb{C})\), \({\mathfrak l}^p(H)\) is the \(p\)th Schatten ideal, is shown (Corollary 3.6). Then cyclic cocycles associated with (bounded) Fredholm modules are computed (Section 3.4). In Chapter 4, first bivariant \(K\)-theory for \(m\)-algebras is exposed according to [1]. The bivariant \(K\)-functor is the universal functor from the given category \({\mathcal C}\) of algebras into an additive category \(D\) that has the properties:
1. It is diffeotopy invariant.
2. It is stable.
3. It is half-exact.
Here, a functor \(E\) is diffeotopy invariant if the evaluation map \(ev_t\), \(t\in [0,1]\), induces an isomorphism \(E(ev_t): E(C^\infty([0, 1],{\mathcal A}))\to E({\mathcal A})\) for any \({\mathcal A}\in{\mathcal C}\), and stable if the canonical inclusion \(\iota:{\mathcal A}\to {\mathcal K}\widehat\otimes{\mathcal A}\) induces an isomorphism for any \({\mathcal A}\in{\mathcal C}\), where \({\mathcal K}\) is the algebra of infinite \(\mathbb{N}\times \mathbb{N}\)-matrices with rapidly decreasing coefficients (Corollary 4.15). The bivariant Chern–Connes character is the unique functor \(ch: kk_0\to HP_0\) such that \[ ch(kk_0(\alpha))= HP_0(\alpha)\in HP_0({\mathcal A},{\mathcal B}),\;\alpha:{\mathcal A}\to {\mathcal B}\text{ a morphism} \] (Corollary 4.17). Regarding \(e\) to be a homomorphism from \(\mathbb{C}\) to \({\mathcal K}\otimes{\mathcal A}\) and viewing it as an element \(h_e\in kk_0(\mathbb{C},{\mathcal A})\), \(ch([e])= ch(h_e)\) holds. It is also stated that the Chern-Connes character gives a transformation \[ ch^{(p)}: kk_*({\mathcal A}, {\mathfrak l}^p\widehat\otimes{\mathcal B}\to HP_*({\mathcal A},{\mathcal B}),\;ch^{(p)}(x\cdot kk(\iota^p))= ch(x) \] (Corollary 4.21), which generalizes previous constructions of the Chern-Connes character in special cases. In the study of discrete subgroups of Lie groups and of quantum field theory, \(\theta\)-summable, but not finitely summable Fredholm modules are needed. To handle such modules, cyclic theories for complete bornological algebras are presented in Chapter 5 [as for bornological vector space, the author refers to H. Hogbe-Nlend, “Bornologies and functional analysis” (North-Holland, Amsterdam) (1977; Zbl 0359.46004)]. If \(A\) is a bornological algebra, the algebra \(\Omega A\) of noncommutative differential forms over \(A\) is a bornological algebra by the bornology generated by \[ \bigcup_{n\geq 0} S(dS)^n\cup (ds)^{n+1}. \] Then, by using completions of \(X\)-complexes, the entire cyclic homology and bivariant entire cyclic homology of bornological algebras \(A\) and \(B\) are defined by \[ HE_*(B)= H_*(X(TB)^c),\quad HE_*(A,B)= H_*({\mathcal L}(X(TA)^c, X(TB)^c)) \] (Definition 5.3). Entire cyclic cohomology can be viewed as a version of periodic cyclic cohomology that contains, in addition, certain infinite-dimensional cycles. The Chern character for \(\theta\)-summable Fredholm modules can be defined by using entire cyclic cohomology [A. Connes, \(K\)-theory 1, 519–548 (1988; Zbl 0657.46049); A. Jaffe, A. Liesniewski and K. Osterwalder, Commun. Math. Phys. 118, 1–14 (1988; Zbl 0656.58048)]. But neither periodic nor entire cohomology yields good results for big algebras like \(C^*\)-algebras. Local cyclic cohomology was defined to obtain reasonable results for \(C^*\)-algebras [M. Puschnigg, Invent. Math. 149, 153–194 (2002; Zbl 1019.22002)]. This theory is explained in Section 5.2, the last section of this chapter.
The second part consists of Section 2: Hochschild and cyclic homology of algebras, Section 3: The cyclic complex \(C^\lambda_\bullet\), Section 4: Noncommutative differential calculus, Section 5: Cyclic objects, Section 6: Examples, Section 7: Index theorems, and Section 8: Riemann-Roch theorem for D-modules. The primary object of this part is the negative cyclic complex \(CC^-_\bullet(A)\). Other complexes are defined as results of natural procedures applied to \(CC^-_*(A)\) in Section 2. Cyclic theory via the Connes complex is presented in Section 3. Both sections concludes to expose how to generalize classical algebraic structures arising in calculus by using these complexes. The noncommutative differential calculus of Tamarkin and the author provides richer algebraic structure on Hochschild chains and cochains than that of obtained in Section 2 and 3 ([2]). It uses the differential graded Gerstenhaber algebra \({\mathcal V}^\bullet(A)\) associated uniquely to \(A\) (Theorem 4.2.1). Then the calculus on \(A\) and the relation between this calculus and noncommutative calculus defined in previous sections are explained (Section 4). The author remarks that this noncommutative differential calculus is constructed using some inexplicit formulas (Section 4.4); a choice of coefficients in these formulas depends on a choice of a Drinfel’d associator [V. G. Drinfel’d, Leningr. J. Math. 2, 829–860 (1991; Zbl 0728.16021)]. In particular, the Grothendieck-Teichmüller group acts on the space of all such calculi. There is another noncommutative calculus developed in [5], [6] which has the renormalization group as a hidden symmetry. The author says that to find a unified framework of the two approaches to noncommutative calculus is an interesting problem. A cyclic object of a category \(C\) is a simplicial object \(X\) together with suitable morphisms. In Section 5, this is explained after dealing with simplicial objects. Then in Section 6, examples of noncommutative calculus, mainly focussed on the applications to deformation quantization, are presented. In Section 7, several index theorems for deformations are reviewed (cf. [3], [4]). The proof of the Riemann-Roch theorem for D-modules (Theorem 8.0.2) was done by reducing it to the index theorem for symplectic deformation [Theorem 7.2.1.; cf. P. Bressler, R. Nest and B. Tsygan, Adv. Math. 167, 1–25 (2002; Zbl 1021.53064) and Adv. Math. 167, 26–73 (2002; Zbl 1021.53065)]. This is reviewed in Section 8.
An important application of cyclic theory that was not exposed in the previous parts, is the local index theorem (transverse index theorem) of Connes-Moscovici ([5], [6]). The third part is the translation of the second author’s exposition of the local index theorem [G. Skandalis, Astérisque 282, 345–364 (2002; Zbl 1035.58007)]. The local index theorem aims to get an index theorem on the noncommutative space \(M/\Gamma\), where \(M\) is a smooth manifold and \(\Gamma\) a countable group of diffeomorphisms of \(M\). It uses a hypoelliptic pseudodifferential operator which is almost invariant by all the diffeomorphism of \(M\). The construction of such an operator is exposed in Section 3. The index of this operator can be computed by using higher analogues of the Wodzicki-Guillemin residue (Section 2). This formula is hard to compute, but it is simplified by using a noncommutative and noncocommutative Hopf algebra \({\mathcal H}_n\) of transverse vector fields on \(\mathbb{R}^n\). This is explained in Section 4, the last section. Necessary preliminaries, such as the Wodzicki-Guillemin residue, cyclic cohomology, and \(p\)-summable cocycles, are summarized in Section 1.