Piotr Nowak

I was going to reference just a couple of Piotr's articles but then I thought that I might as well point to his entire home page. Everything is good here! In particular he shows that coarse embeddability into a Hilbert space (or into ell-one) is not the same as property A. The example is devastatingly simple: take the disjoint union of n-fold products of copies of some finite group (e.g. the group of order 2). Notice that the spaces here are quasi-isometric to cubes in R^n with the ell-one metric. If one took the ell-two (Euclidean) metric instead, wouldn't one get Yu's old counterexample to coarse Baum-Connes? Something interesting seems to be going on here.

[Of course these aren't bg spaces. Is there a bg space with the same property?]

## Sunday, November 13, 2005

## Monday, November 07, 2005

### [math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

Authors: Severino T. Melo, Cintia C. Silva

Subj-class: Operator Algebras; K-Theory and Homology

MSC-class: 46L80; 47G30

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2\pi-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol \sigma and an operator-valued symbol \gamma. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma. For the second computation, we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

Authors: Severino T. Melo, Cintia C. Silva

Subj-class: Operator Algebras; K-Theory and Homology

MSC-class: 46L80; 47G30

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2\pi-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol \sigma and an operator-valued symbol \gamma. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma. For the second computation, we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

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