Abstract
We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to free semigroup algebras as well.
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Badea C. (1998). The stable rank of topological algebras and a problem of R.G. Swan. J. Funct. Anal. 160: 42–78
Bass, H.: K-theory and stable algebra, vol. 22, pp. 489–544. Publications Mathématiques. Institut des Hautes Études Scientifiques, Paris (1964)
Bunce J.W. (1984). Models for n-tuples of noncommuting operators. J. Funct. Anal. 57: 21–30
Caradus S.R., Pfaffenberger W.E. and Yood B. (1974). Calkin algebras and algebras of operators on banach spaces, Lecture Notes in Pure Applied Mathematics, vol. 9. Marcel Dekker, Inc., New York
Davidson K.R. (1988). Nest algebras. Triangular forms for operators on a Hilbert space, Pitman Research Notes in Mathematics, vol. 191. Longman Scientific and Technical, Harlow
Davidson K.R., Harrison K.J. and Orr J.L. (1995). Epimorphisms of nest algebras. Int. J. Math. 6: 657–687
Davidson K.R., Katsoulis E.G. and Pitts D.R. (2001). The structure of free semigroup algebras. J. Reine Angew. Math. 533: 99–125
Davidson K.R. and Pitts D.R. (1999). Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. Lond. Math. Soc. 78: 401–430
Herman R.H. and Vaserstein L.N. (1984). The stable range of C *-algebras. Invent. Math. 77: 553–555
Herrero D.A. (1989). Approximation of Hilbert space operators I, Pitman Research Notes in Mathematics, vol. 224, 2nd edn. Longman Scientific and Technical, Harlow
Jones P.W., Marshall D. and Wolff T. (1986). Stable rank of the disc algebra. Proc. Am. Math. Soc. 96: 603–604
Popescu G. (1991). Von Neumann inequality for \(({\mathcal{B}}({\mathcal{H}})^n)_1\). Math. Scand. 68: 292–304
Popescu G. (1996). Non-commutative disc algebras and their representations. Proc. Am. Math. Soc. 124: 2137–2148
Rieffel M. (1983). Dimension and stable rank in the K-theory of C *-algebras. Proc. Lond. Math. Soc. 46: 301–333
Vaserstein L.N. (1971). Stable rank of rings and dimesionality of topological spaces. Funct. Anal. Appl. 5: 102–110
Warfield Jr. R.B. (1980). Cancellation of modules and groups and stable range of endomorphism rings. Pac. J. Math. 91: 457–485
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K. R. Davidson, L. W. Marcoux, H. Radjavi’s research was supported in part by NSERC (Canada).
An erratum to this article can be found at http://dx.doi.org/10.1007/s00208-008-0229-0
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Davidson, K.R., Levene, R.H., Marcoux, L.W. et al. On the topological stable rank of non-selfadjoint operator algebras. Math. Ann. 341, 239–253 (2008). https://doi.org/10.1007/s00208-007-0180-5
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DOI: https://doi.org/10.1007/s00208-007-0180-5