Deformations, asymptotic morphisms and bivariant \(K\)-theory. (Déformations, morphismes asymptotiques et \(K\)-théorie bivariante.) (French) Zbl 0717.46062
For two \(C^*\)-algebras \(A\) and \(B\) the authors consider two equivalent data: of a deformation of \(A\) into \(B\), and of an asymptotic morphism from \(A\) into \(B\). A deformation is considered as a continuous field \((A(t),\Gamma\)) of \(C^*\)-algebras over \([0,1]\) such that \(A(0)=A\), \(A(t)=B\), \(\forall t\neq 0\). Following the definition of a continuous field of \(C^*\)-algebras, one can choose for each \(a\in A\) a section \(\alpha\in \Gamma\), passing through \(a\), i.e. \(\alpha (0)=a\). Such a choice led us to construct a family \((\phi_ t)_{t\in [1,\infty)}\) of maps from \(A\) into \(B\), \(\phi_ t(a):=\alpha (a)(1/t)\in B\), satisfying the two conditions what follow:
(1) For each \(a\in A\), the map \(t\mapsto \phi_ t(a)\) is continuous (in the norm topology).
(2) For each \(a,a'\in A\), \(\lambda\in\mathbb C\) one has the following limits in the norm topology \[ \lim_{t\to \infty}(\phi_ t(a)+\lambda \phi_ t(a')-\phi_ t(a+\lambda a'))=0,\quad\lim_{t\to \infty}(\phi_ t(a)\phi_ t(a')-\phi_ t(aa'))=0, \]
\[ \lim_{t\to \infty}(\phi_ t(a)^*-\phi_ t(a^*))=0. \] Any (abstract) family \((\phi_ t)_{t\in [1,\infty)}\) of maps from \(A\) into \(B\) satisfying these two conditions is called an asymptotic morphism from \(A\) into \(B\).
An easy modification gives us the composition, the homotopy and the tensor product of two asymptotic morphisms. These notions are easily extended to the set \([[A,B]]\) of homotopy classes. The additive category \(E(A,B)\) is defined as \[ E(A,B):=[[SA\otimes {\mathcal K},\, SB\otimes {\mathcal K}]], \] where \(SA:=C_ 0({\mathbb R})\otimes A\), \(C_ 0({\mathbb R})\) is the well-known algebra, and \({\mathcal K}\) is the algebra of compact operators in a separable Hilbert space.
The main result of the work under review asserts (Theorem 7), that the bivariant functor \(E(A,B)\) is stable (i.e. unchanged when \(A\) is replaced by \(A\otimes {\mathcal K})\) half exact and universal. It is therefore easy to deduce that the obtained theory provides a new realization of the Kasparov’s theory \(KK(A,B)\) (Corollary 9).
(1) For each \(a\in A\), the map \(t\mapsto \phi_ t(a)\) is continuous (in the norm topology).
(2) For each \(a,a'\in A\), \(\lambda\in\mathbb C\) one has the following limits in the norm topology \[ \lim_{t\to \infty}(\phi_ t(a)+\lambda \phi_ t(a')-\phi_ t(a+\lambda a'))=0,\quad\lim_{t\to \infty}(\phi_ t(a)\phi_ t(a')-\phi_ t(aa'))=0, \]
\[ \lim_{t\to \infty}(\phi_ t(a)^*-\phi_ t(a^*))=0. \] Any (abstract) family \((\phi_ t)_{t\in [1,\infty)}\) of maps from \(A\) into \(B\) satisfying these two conditions is called an asymptotic morphism from \(A\) into \(B\).
An easy modification gives us the composition, the homotopy and the tensor product of two asymptotic morphisms. These notions are easily extended to the set \([[A,B]]\) of homotopy classes. The additive category \(E(A,B)\) is defined as \[ E(A,B):=[[SA\otimes {\mathcal K},\, SB\otimes {\mathcal K}]], \] where \(SA:=C_ 0({\mathbb R})\otimes A\), \(C_ 0({\mathbb R})\) is the well-known algebra, and \({\mathcal K}\) is the algebra of compact operators in a separable Hilbert space.
The main result of the work under review asserts (Theorem 7), that the bivariant functor \(E(A,B)\) is stable (i.e. unchanged when \(A\) is replaced by \(A\otimes {\mathcal K})\) half exact and universal. It is therefore easy to deduce that the obtained theory provides a new realization of the Kasparov’s theory \(KK(A,B)\) (Corollary 9).
Reviewer: Do Ngoc Diep (Hà Nội)
MSC:
46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
19K35 | Kasparov theory (\(KK\)-theory) |