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Relative \(K\) homology and \(C^*\) algebras. (English) Zbl 0755.46035
Let \(A\) be a separable nuclear \(C^*\) algebra with unit and let \({\mathcal J}\) be a closed two-sided ideal in \(A\). The relative \(K\)-homology group \(K^ 0(A,{\mathcal J})\) is defined as the set of homotopy-isomorphic equivalence classes of the objects of type \(({\mathcal H}_ 0,\psi_ 0,{\mathcal H}_ 1,\psi_ 1,T)\) such that \(({\mathcal H}_ i,\psi_ i)\), \(i=1,2\), are \(*\)-representations of \(A\) in \({\mathcal H}_ i\), and \(T\) is a partial isometry intertwining (modulo compact operators).
The authors proved some isomorphisms between these groups and the corresponding Kasparov groups, namely \[ K^ 0(A)\cong KK^ 0(A,\mathbb{C}), \qquad K^ 0(A,{\mathcal J})\cong KK^ 0({\mathcal J},\mathbb{C}), \qquad K^ 1(A)\cong KK^ 1(A,\mathbb{C}) \] and the five term exact sequence \[ K^ 0(A/{\mathcal J})\to K^ 0(A)\to K^ 0(A;{\mathcal J})@>\partial>>K^ 1(A/{\mathcal J})\to K^ 1(A). \] For topological spaces, this theory is referred as analytic \(K\) homology, \[ K_ *^ a(X):=K^*(C(X)). \] The theory can be applied in this cases to obtain various index formulas, including one for Toeplitz operators. In the previous work [P. Baum, R. G. Douglas and M. E. Taylor, J. Diff. Geom. 30, No. 3, 761-804 (1989; Zbl 0697.58050)], the authors and Taylor considered also a topological setting of \(K_ *^ t(X)\) and there were a natural isomorphism \[ K_ *^ t(X)\cong K_ *^ a(X). \] The final two sections of the paper are devoted to the topological relative \(K_ *^ t(X,Y)\) and topological \(K\) homology with proper support \(\hat K_ *(X)\) and existence of coercive boundary conditions for elliptic differential operators.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K33 Ext and \(K\)-homology
19K35 Kasparov theory (\(KK\)-theory)
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