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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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