Finite and torsion KK-theories

📝 Abstract

We develop a finite KKG-theory of C*-algebras following Arlettaz- H.Inassaridze’s approach to finite algebraic K-theory. The Browder- Karoubi-Lambre’s theorem on the orders of the elements for finite algebraic K-theory is extended to finite KKG-theory. A new bivariant theory, called torsion KK-theory is defined as the direct limit of finite KK-theories. Such bivariant K-theory has almost all KKG-theory properties and one has a long exact sequence relating KK-theory, rational bivariant K-theory and torsion KK-theory. For a given homology theory on the category of separable GC*-algebras finite, rational and torsion homology theories are introduced and investigated. In particular, we formulate finite, torsion and rational versions of Baum-Connes Conjecture. The later is equivalent to the investigation of rational and q-finite analogues for Baum-Connes Conjecture for all prime q.

💡 Analysis

We develop a finite KKG-theory of C*-algebras following Arlettaz- H.Inassaridze’s approach to finite algebraic K-theory. The Browder- Karoubi-Lambre’s theorem on the orders of the elements for finite algebraic K-theory is extended to finite KKG-theory. A new bivariant theory, called torsion KK-theory is defined as the direct limit of finite KK-theories. Such bivariant K-theory has almost all KKG-theory properties and one has a long exact sequence relating KK-theory, rational bivariant K-theory and torsion KK-theory. For a given homology theory on the category of separable GC*-algebras finite, rational and torsion homology theories are introduced and investigated. In particular, we formulate finite, torsion and rational versions of Baum-Connes Conjecture. The later is equivalent to the investigation of rational and q-finite analogues for Baum-Connes Conjecture for all prime q.

📄 Content

arXiv:0909.5514v1 [math.KT] 30 Sep 2009 FINITE AND TORSION KK-THEORIES HVEDRI INASSARIDZE AND TAMAZ KANDELAKI Abstract. We develop a finite KKG-theory of C∗-algebras following Arlettaz- H.Inassaridze’s approach to finite algebraic K-theory [1] . The Browder- Karoubi-Lambre’s theorem on the orders of the elements for finite algebraic K-theory [ , ] is extended to finite KKG-theory. A new bivariant theory, called torsion KK-theory is defined as the direct limit of finite KK-theories. Such bivariant K-theory has almost all KKG-theory properties and one has the following exact sequence · · · → KKG n (A, B) → KKG n (A, B; Q) → KKG n (A, B; T) → · · · relating KK-theory, rational bivariant K-theory and torsion KK-theory. For a given homology theory on the category of separable GC∗-algebras finite, rational and torsion homology theories are introduced and investigated. In particular, we formulate finite, torsion and rational versions of Baum-Connes Conjecture. The later is equivalent to the investigation of rational and q-finite analogues for Baum-Connes Conjecture for all prime q. Introduction In this paper we provide a new bivariant theory, which will be called torsion equivariant KKG-theory. That is closely connected with the usual and rational versions of KKG-theories. By definition torsion KKG-theory is a direct limit of KKG-theory with coefficients in Zq (q-finite KKG-theory in our terminology), where q runs over all natural numbers ≥2. This new bivariant homology theory has all the properties of KKG-theory except of the existence of the identity morphism. We arrive to the following principle: some of problems that arise in usual KKG- theory may be reduced to suitable problems in rational, finite and torsion KKG- theories. Namely, it will be shown that Baum-Connes conjecture has analogues in finite, torsion and rational KK-theories and Baum-Connes assembly map is an isomorphism if and only if its rational and finite assembly maps are isomorphisms for all prime q (Theorem 3.5). As a technical tool, we mainly work with homology theories on the category of C∗-algebras with action of a fix locally compact group G. In sections 1 and 2 for a given homology theory H torsion and q-finite homology theories H(q) are con- structed and their properties are investigated. Much of these properties are known for experts in some concrete form, but we could not find suitable references for our purposes. They are redefined and reinvestigated here. Furthermore, in section 2 we define and investigate a new homology theory, so called torsion homology theory. Especially, we make accent on the following twosided long exact sequence of abelian The authors were partially supported by GNSF-grant, INTAS-Caucasus grant and Volkswagen Foundation grant. 1 2 HVEDRI INASSARIDZE AND TAMAZ KANDELAKI groups · · · →HT n+1(A) →Hn(A) r−→Hn(A) ⊗Q →HT n(A) →Hn−1(A) r−→· · · for any GC∗-algebra A which is used concretely for bivariant KK-theories in the sequel section. In particular, based on results of these sections we list properties of torsion and finite bivariant KKG-theories. Besides, there exists a long exact sequence, which is similar to the above long exact sequence: (0.1) · · · →KKG n+1(A, B; Q) →KKG n+1(A, B; Q/Z) → →KKG n (A, B) Ratn −−−→KKG n (A, B; Q) →KKG n (A, B; Q/Z)) →· · · The similar result for K-theory of bornological algebras one can find in [5]. The rational bivariant KK-theory and the torsion bivariant KK-theory have all the properties of usual bivariant KK-theory. The only difference is that the torsion case hasn’t unital morphisms. Note that rational bivariant KK-theory used in this paper differs from the similar one defined in [3]. In the next section 3 we study torsion and q-finite KK-theories, where the fi- nite KK-theory is redefined following Arlettaz-H.Inassaridze’s approach to finite algebraic K-theory [1]. Sections 4 and 5 are devoted to the proof of the following Browder-Karoubi-Lambre’ theorem for finite KK-theory (see Theorem 5.5): Let A and B be, respectively, separable and σ-unital C∗-algebras, real or com- plex; and G be a metrizable compact group. Then, for all integer n, (1) q · KKG n (A, B; Z/q) = 0, if q −2 is not divided by 4; (2) 2q · KKG n (A, B; Z/q) = 0, if 4 divides q −2. It is clear that this result holds for non-unital rings too. For finite algebraic K-theory this theorem for n = 1 was proved algebraically by Karoubi and Lambre [], and for n > 1 by Browder [2]. The key idea to carry out this problem is its reduction to the algebraic K-theory case. This is realized by two steps. First we calculate finite topological K-theory of C∗-algebras and additive C∗-categories by finite algebraic K-theory of rings. Then generalizing the main result of [8], finite bivariant KKG-theory is calculated by finite topological K-theory of the additive C∗-category of Fredholm modules. When G is a locally compact group, it is more complicated to get the similar result for finite G-equivariant bivariant KK-theory and we intend to investigate this problem

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