# C-algebras and elliptic theory by Bogdan Bojarski, Alexander S. Mishchenko, Evgenij V.

By Bogdan Bojarski, Alexander S. Mishchenko, Evgenij V. Troitsky, Andrzej Weber, Dan Burghelea, Richard Melrose, Victor Nistor

This quantity includes the lawsuits of the convention on "C*-algebras and Elliptic idea" held in Bedlewo, Poland, in February 2004. It comprises unique study papers and expository articles focussing on index conception and topology of manifolds.

The assortment deals a cross-section of important fresh advances in different fields, the most topic being K-theory (of C*-algebras, equivariant K-theory). a couple of papers is expounded to the index conception of pseudodifferential operators on singular manifolds (with obstacles, corners) or open manifolds. additional subject matters are Hopf cyclic cohomology, geometry of foliations, residue concept, Fredholm pairs and others. The extensive spectrum of topics displays the various instructions of analysis emanating from the Atiyah-Singer index theorem.

Contributors:

B. Bojarski, J. Brodzki, D. Burghelea, A. Connes, J. Eichhorn, T. Fack, S. Haller, Yu.A. Kordyukov, V. Manuilov, V. Nazaikinskii, G.A. Niblo, F. Nicola, I.M. Nikonov, V. Nistor, L. Rodino, A. Savin, V.V. Sharko, G.I. Sharygin, B. Sternin, ok. Thomsen, E.V. Troitsky, E. Vasseli, A. Weber

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Given that (Tn ⊗ idQ )(idA ⊗ q)(x) = (idA ⊗ q)(Tn ⊗ idB )(x) we have that (idA ⊗ q)(Tn ⊗ idB )(x) = 0. Since every operator Tn is of ﬁnite rank, (Tn ⊗ idB )(x) belongs to the algebraic tensor product of A B of the algebras A and B. The algebraic tensor product is an exact functor, so the vanishing condition (idA ⊗ q)(Tn ⊗ idB )(x) = 0 implies that (Tn ⊗ idB )(x) is an element of A ⊗ I. Therefore x = lim(Tn ⊗ idB )(x) is also in A ⊗ I. This proves that the kernel of the map idA ⊗ q is identical to A ⊗ I for any algebras B and I, which implies that A is exact.

Considering Euler chains with real coeﬃcients one obtains in exactly the same way an aﬃne version of H1 (M ; R) which we will denote by Eulx0 (M ; R). There is an obvious map Eulx0 (M ; Z) → Eulx0 (M ; R) which is aﬃne over the homomorphism H1 (M ; Z) → H1 (M ; R). Remark 3. Another way to understand the H1 (M ; Z) action on Eulx0 (M ; Z) is the following. Suppose n > 2 and represent [σ] ∈ H1 (M ; Z) by a simple closed curve σ. Choose a tubular neighborhood N of S 1 considered as vector bundle N → S 1 .

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