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Let A be a C-algebra and a-endomorphism of A. The analogue of Pimsner-Voiculescu exact sequences are obtained for the pair (A; ). We prove that the corresponding Toeplitz algebra remains KK-equivalent to A.We also consider the situation where a semigroup ( t )t2R+ of-endomorphisms is acting on A and formulate similar exact sequences. In this part we use the language of Connes-Higson E-theory.

Toeplitz algebras associated with endomorphisms and Pimsner–Voiculescu exact sequences