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Mapping cones and exact sequences in KK-theory. (English) Zbl 0603.46063
Let A and B be \(C^*\)-algebras and \(\phi: A\to B\) a *-homomorphism. The mapping cone for \(\phi\) is \(C_{\phi}=\{(x,f)\in A\oplus B[0,1):(x)=f(0)\}\) where \(B[0,1)\) denotes the \(C^*\)-algebra of B-valued continuous functions on [0,1) vanishing at infinity. Let \(p: C_{\phi}\to A\) and \(i: B[0,1)\to C_{\phi}\) be given by \(p(x,f)=x\) and \(i(f)=(0,f)\). The authors prove that, for A,B,C,D graded \(C^*\)-algebras and \(\phi\) : \(A\to B\) a grading preserving homomorphism, the ”Puppe sequences” \[ KK(D,A[0,1))\to KK(D,B[0,1))\to KK(D,C_{\phi})\to KK(D,A)\to KK(D,B) \] and, if A and B are separable, \[ KK(B,D)\to KK(A,D)\to KK(C_{\phi}D)\to KK(B[0,1),D)\to KK(A[0,1),D) \] are exact. Using this theorem and an excision type result the existence of the long exact sequence associated with an ideal, due to G. G. Kasparov [Math. USSR, Izv. 16, 513-572 (1981; Zbl 0464.46054), § 7] is proved. Two notions of cobordism and cohomotopy for Kasparov bimodules are introduced and it is proved that, if A is separable, two Kasparov (A,B)- bimodules are cobordant if and only if they are homotopic (cohomotopic). Finally, the authors construct a ”dual Puppe sequence” \[ KK(B\otimes \hat S,D)\to KK(A\otimes \hat S,D)\to KK(\hat C_{\phi},D)\to KK(B,D)\to KK(A,D) \] where D is any graded \(C^*\)-algebra, A and B are separable and \(\hat C=Ker \lambda\) where \({\mathfrak S}\) is the Toeplitz algebra \(C^*(v)\) for a non unitary isometry v, \(\lambda\) : \({\mathfrak S}\to {\mathbb{C}}\) is the *-homomorphism given by \(\lambda (v)=1\) and \(\hat S=\hat C/{\mathcal K}\) where \({\mathcal K}\) is the algebra of compact operators, which is contained in \(\hat C.\)
Reviewer: G.Corach

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L05 General theory of \(C^*\)-algebras