# Journal of Operator Theory

Volume 42, Issue 1, Summer 1999 pp. 3-36.

On ${\bbb Z}/2{\bbb Z}$-graded KK-theory and its relation with the graded Ext-functor**Authors**: Ulrich Haag

**Author institution:**Inst. fur Reine Mathematik, Humboldt Universitat Berlin, Ziegelstr. 13A, 10099 Berlin, Germany

**Summary:**This paper studies the relation between ${\rm KK}$-theory and the ${\rm Ext}$-functor of Kasparov for ${\bbb Z}_2$-grade d $C^*$-algebras. We use an approach similar to the picture of J. Cuntz in the ungraded case. We show that the graded ${\rm Ext}$-functor coincides with ${\bbb Z}_2$-equivariant ${\rm KK}$-theory up to a shift in dimen sion and that the graded ${\rm KK}$-functor can be expressed in terms of ${\bbb Z}_2$-equ ivariant ${\rm KK}$-theory. We derive a (double) exact sequence relating both theories.

**Keywords:**$\bbb Z_2$-graded $C^*$-algebras, superquasimorphism, Cuntz picture, universal algebras, double exact sequence

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