Abstract
In this paper, we will define several new isomorphism invariants for C*-algebras by hyponormal partial isometries and discuss the relation between these invariants and K-theory of C*-algebras. This study was in part inspired by the work of H. Lin and H. Su in the context of \({A\mathcal{T}}\)-algebras. An \({{\rm A}\mathcal{T}}\)-algebra often becomes an extension of an \({{\rm A}\mathbb{T}}\)-algebra by an AF-algebra. We show that there is an essential extension of a simple \({{\rm A}\mathbb{T}}\)-algebra which has real rank zero by an AF-algebra such that it has real rank zero and is not an \({A\mathcal{T}}\)-algebra.
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References
Blackadar B.: K-Theory for Operator Algebras. Springer-Verlag, New York (1986)
L. G. Brown, R. Douglas, and P. Fillmore, Unitary equivalence modulo the compact operators and extensions of C*-algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973), pp. 58–128. Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973.
Brown L. G., Douglas R., Fillmore P.: Extensions of C*-algebras and K-homology. Ann. of Math. 105, 265–324 (1977)
Dădărlat M. et al.: Reduction of topological stable rank in inductive limits of C*-algebras. Pacific J. Math. 153, 267–276 (1992)
Elliott G.A.: On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38, 29–44 (1976)
Elliott G.A.: On the classification of C*-algebras of real rank zero. J. Reine Angew. Math. 443, 179–219 (1993)
G. A. Elliott, D. E. Evans, and H. Su, Classification of inductive limits of the Toeplitz algebra tensored with \({\mathcal{K}}\), Operator algebras and quantum field theory (Rome, 1996), 36–50, Internat. Press, Cambridge, MA, 1997.
H. Lin, C*-algebra extensions of C(X), Mem. Amer. Math. Soc. 115 no. 550, (1995)
Lin H.: On the classification of C*-algebras of real rank zero with zero K 1. J. Operator Theory 35, 147–178 (1996)
H. Lin, A classification theorem for infinite Toeplitz algebras, Operator algebras and operator theory (Shanghai, 1997), 219–275, Contemp. Math., 228, Amer. Math. Soc., Providence, RI, 1998.
H. Lin, An Introduction to the Classification of Amenable C*-Algebras, World Scientific, New Jersey/London/Singapore/Hong Kong, 2001.
Lin H., Rørdam M.: Extensions of inductive limits of circle algebras. J. London Math. Soc. 51, 603–613 (1995)
Lin H., Su H.: Classification of direct limits of generalized Toeplitz algebras. Pacific J. Math. 181, 89–140 (1997)
Restorff G., Ruiz E.: On Rørdam’s classification of certain C*-algebras with one nontrivial ideal II. Math. Scand. 101, 280–292 (2007)
Rørdam M.: Classification of extensions of certain C*-algebras by their six term exact sequences in K-Theory. Math. Ann. 308, 93–117 (1997)
Yao H.: On extensions of stably finite C*-algebras. J. Operator Theory 67, 329–334 (2012)
Yao H., Fang X.: Partial isometries and an invariant of C*-algebras. Acta Math. Sinica 27, 799–806 (2011)
Yao H., Fang X.: Notes on \({{\rm A}\mathcal{T}}\)-algebras and extensions of \({{\rm A}\mathbb{T}}\)-algebras by \({\mathcal{K}}\). Bull. London Math. Soc. 42, 275–281 (2010)
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This article is supported by the National Natural Science Foundation of China (No. 11001131) and the NUST Research Funding (No. 2010ZYTS068).
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Yao, H., Fang, X. Partial isometries and extensions of \({{\rm A}{\mathbb T}}\)-algebras. Arch. Math. 99, 137–146 (2012). https://doi.org/10.1007/s00013-012-0417-8
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DOI: https://doi.org/10.1007/s00013-012-0417-8