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Berg's Technique for Pseudo-Actions With Applications to af Embeddings

Published online by Cambridge University Press:  20 November 2018

Terry A. Loring
Terry A. Loring*
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia B3H 3J5
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Abstract

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Berg's interchange technique is generalized to the context of certain new objects called pseudo-actions. This is used to find a more geometric proof of the Pimsner-Voiculescu theorem on the AF embedding of the irrational rotation algebras. Connections with Berg's original results are briefly examined.

Embedding diagrams are introduced to provide a uniform way of describing embeddings of transformation group C*-algebras C(X) ⋊ ℤ into AF algebras. Pimsner has classified the transformation group C* -algebras which can be AF embedded. We present a new proof of this result using embedding diagrams and pseudo-actions. The need to calculate the join of an open cover with its iterates under the transformation has been eliminated.

Keywords

46L0547A5554H20Berg's techniqueAF algebraAF embeddingtransformation group C*-algebra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 1 , 01 February 1991 , pp. 119 - 157
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Berg, I.D., On approximation of normal operators by weighted shifts, Michigan Math. J. 21(1974), 377- 383.Google Scholar
2. Berg, D., On operators which almost commute with the shift, J. Operator Theory 11(1984), 365–377.Google Scholar
3. Berg, I.D. and Davidson, K.R., Almost commuting matrices and a quantitative version of the Brown- Douglas-Fillmore theorem, preprint.Google Scholar
4. Davidson, K.R., Berg's technique and irrational rotation algebras, Proc. R. Ir. Acad. 84A(1984), 117–123.Google Scholar
5. Exel, R. and Loring, T.A.. Extending cellular homology to C*-algebras, preprint.Google Scholar
6. Loring, T.A., The K-theory of AF embeddings of the rational rotation algebras, A'-theory, to appear.Google Scholar
7. Pimsner, M.V., Embedding some transformation group C* -algebras into AF algebras, Ergod. Th. & Dynam. Sys. 3(1983), 613–626.Google Scholar
8. Pimsner, M.V. and Voiculescu, D., Imbedding the irrational rotation algebras into an AF-algebra, J. Operator Theory 4(1980), 201–210.Google Scholar
9. Putnam, I.F., The C*-algebras associated to minimal homeomorphisms of the Cantor set, Pacific J. Math. 136(1989), 329–354.Google Scholar
10. Rieffel, M.A., C*-algebras associated with irrational rotations, Pacific J. Math. 93(1981), 415–429.Google Scholar
11. Veršik, A.M., Uniform algebraic approximation of shift and multiplication operators, Soviet Math. Dokl. 24(1981), 97–100.Google Scholar