Semiprojectivity of universal -algebras generated by algebraic elements
Author:
Tatiana Shulman
Journal:
Proc. Amer. Math. Soc. 140 (2012), 1363-1370
MSC (2010):
Primary 46L05, 46L35
Published electronically:
August 9, 2011
MathSciNet review:
2869120
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Abstract: Let be a polynomial in one variable whose roots all have multiplicity more than 1. It is shown that the universal -algebra of a relation , , is semiprojective and residually finite-dimensional. Applications to polynomially compact operators are given.
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Bibliography
1. C. A. Akemann and G. K. Pedersen, Ideal perturbations of elements in -algebras, Math. Scand. 41 (1977), 117-139. MR 0473848 (57:13507)
2. B. Blackadar, Shape theory for -algebras, Math. Scand. 56 (1985), 249 - 275. MR 813640 (87b:46074)
3. C. K. Chui, D. A. Legg, P. W. Smith, and J. D. Ward, On a question of Olsen concerning compact perturbations of operators, Michigan Math. J. 24, 1 (1977), 119-127. MR 0451005 (56:9295)
4. D. Hadwin, Lifting algebraic elements in -algebras, J. Funct. Anal. 127 (1995), 431-437. MR 1317724 (95m:46092)
5. T. A. Loring, Lifting solutions to perturbing problems in -algebras, volume 8 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1997. MR 1420863 (98a:46090)
6. T. A. Loring, G. K. Pedersen, Smoothing techniques in -algebra theory, J. Operator Theory 37 (1997), 3-21. MR 1438197 (98c:46122)
7. G. J. Murphy, -algebras and operator theory, Academic Press, Boston-New York, 1990. MR 1074574 (91m:46084)
8. C. L. Olsen and G. K. Pedersen, Corona -algebras and their applications to lifting problems, Math. Scand. 64 (1989), 63-86. MR 1036429 (91g:46064)
9. C. L. Olsen, A structure theorem for polynomially compact operators, Amer. J. Math. 93 (1971), 686-698. MR 0405152 (53:8947)
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11. C. L. Olsen, Norms of compact perturbations of operators, Pacific J. Math. 68, 1 (1977), 209-228. MR 0451010 (56:9300)
12. G. K. Pedersen, -algebras and Their Automorphism Groups, LMS Monographs 14, Academic Press, London-New York, 1979. MR 548006 (81e:46037)
13. T. Shulman, Lifting of nilpotent contractions, Bull. London Math. Soc. 40, 6 (2008), 1002-1006. MR 2471949 (2009k:46102)
14. R. R. Smith and J. D. Ward, A note on polynomial operator approximation, Proc. Amer. Math. Soc. 88, 3 (1983), 491-494. MR 699420 (84j:47008)
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Additional Information
Tatiana Shulman
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitet- sparken 5, DK-2100 Copenhagen, Denmark
Email:
shulman@math.ku.dk, tatiana_shulman@yahoo.com
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11144-4
Keywords:
Projective and semiprojective $C^{*}$-algebras,
stable relation,
lifting problem,
$M$-ideal
Received by editor(s):
June 13, 2009
Received by editor(s) in revised form:
January 5, 2011
Published electronically:
August 9, 2011
Communicated by:
Marius Junge
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.