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Factorization of Positive Invertible Operators in af Algebras

Published online by Cambridge University Press:  20 November 2018

Houben Huang  and
Timothy D. Hudson
Houben Huang
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1
Timothy D. Hudson
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, Ontario N2L 3G1
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Abstract

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We examine the problem of factoring a positive invertible operator in an AF C*-algebra as T*T for some invertible operator T with both T and T-1 in a triangular AF subalgebra. A factorization theorem for a certain class of positive invertible operators in AF algebras is proven. However, we explicitly construct a positive invertible operator in the CAR algebra which cannot be factored with respect to the 2∞ refinement algebra. Our main result generalizes this example, showing that in any AF algebra, there exist positive invertible operators which fail to factor with respect to a given triangular AF subalgebra. We also show that in the context of AF algebras, the notions of having a factorization and having a weak factorization are the same.

Keywords

46L0547D25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 421 - 435
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Arveson, W.B., Interpolation problems in nest algebras, J. Funct. Anal. 20(1975), 208–233.Google Scholar
2. Baker, R.L., Triangular UHF algebras, J. Funct. Anal. 91(1990), 182–212.Google Scholar
3. Bratteli, O., Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc. 171(1972), 195–234.Google Scholar
4. Davidson, K.R., Nest Algebras, Pitman Res. Notes Math. 191, Longman Sci. Tech., London, 1988.Google Scholar
5. Davidson, K.R. and Huang, H., Universal factorization of positive operators, Indiana Univ. Mtith ‘., to appear.Google Scholar
6. Donsig, A.P., Semisimple triangular AF algebras, J. Funct. Anal. 111(1993), 323–349.Google Scholar
7. Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc. 95(1960), 318–340.Google Scholar
8. Gohberg, I.C. and Krein, M.G., Factorization of operators in Hilbert space, Acta Sci. Math. (Szeged) 25(1964), 90–123. English transi., Amer. Math. Soc. Transi. Ser. 2 51(1966), 155–188.Google Scholar
9. Gohberg, I.C., Theory and applications ofVolterra operators in Hilbert space, ‘Nauka', Moscow, 1967. English transi., Transi. Math. Monographs 24, Amer. Math. Soc, Providence, Rhode Island, 1970.Google Scholar
10. Hopenwasser, A. and Power, S.C., Classification of limits of triangular matrix algebras, Proc. Edinburgh Math. Soc. 36(1992), 107–121.Google Scholar
11. Huang, H., Factorization of Positive Invertible Operators, Ph.D. Dissertation, University of Waterloo, August, 1993.Google Scholar
12. Hudson, T.D., Ideals in triangular AF algebras, Proc. London Math. Soc, to appear.Google Scholar
13. Larson, D.R., Nest algebras and similarity transformations, Ann. of Math. 121( 1985), 409–427.Google Scholar
14. Muhly, P.S. and Baruch Solel, Subalgebras ofgroupoid C*-algebras, J. Reine Angew. Math. 402(1989), 41–75.Google Scholar
15. Muhly, P.S., Saito, K-S. and Solel, B., Coordinates for triangular operator algebras, Ann. of Math. 127(1988), 245–278.Google Scholar
16. Muhly, P.S., Coordinates for triangular operator algebras II, Pacific J. Math. 137(1989), 335–369.Google Scholar
17. Orr, J.L. and Peters, J.R., Some representations of TAF algebras, preprint, 1992.Google Scholar
18. Peters, J.R., Poon, Y.T. and Wagner, B.H., Triangular AF algebras, J. Operator Theory 23(1990), 81–114.Google Scholar
19. Peters, J.R., Analytic TAF algebras, Canad. J. Math. 45(1993), 1009–1031.Google Scholar
20. Peters, J.R. and Wagner, B.H., Triangular AF algebras and nest subalgebras o/UHF algebras, J. Operator Theory 25(1991), 79–123.Google Scholar
21. Peters, J.R. and Wogen, W.R., Reflexive subalgebras of AF algebras, preprint, 1992.Google Scholar
22. Pitts, D.R., Factorization problems for nests: Factorization methods and characterizations of the universal factorization property, J. Funct. Anal. 79(1988), 57–90.Google Scholar
23. Power, S.C., Limit Algebras: An introduction to subalgebras of C*-algebras, Pitman Res. Notes Math. 278, Longman Sci. Tech., London, 1992.Google Scholar
24. Power, S.C., Factorization in analytic operator algebras, J. Funct. Anal. 67(1986), 413–432.Google Scholar
25. Power, S.C., Nuclear operators in nest algebras, J. Operator Theory 10(1983), 146–148.Google Scholar
26. Power, S.C., On ideals of nest subalgebras of C*-algebras, Proc. London Math. Soc. 50(1985), 314–332.Google Scholar
27. Power, S.C., Classification of tensor products of triangular operator algebras, Proc. London Math. Soc. 61(1990), 571–614.Google Scholar
28. Power, S.C., The classification of triangular subalgebras of AF C*-algebras, Bull. London Math. Soc. 22(1990), 269–272.Google Scholar
29. Sun, J.G., Perturbation bounds for the Cholesky and QR factorizations, BIT 31(1991), 341–352.Google Scholar
30. Ventura, B.A., Strongly maximal triangular AF algebras, Internat. J. Math. 2(1991), 567–598.Google Scholar