Normal operators in -algebras

without nice approximants

Authors:
Don Hadwin and Terry A. Loring

Journal:
Proc. Amer. Math. Soc. **125** (1997), 159-161

MSC (1991):
Primary 46L80, 47A58; Secondary 46L05, 47C15

MathSciNet review:
1371125

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The second author constructed a separable direct limit -algebra with real rank zero containing a normal element whose spectrum is the closed unit disk that is not the limit of normal elements in the limiting algebras, and is not a limit of normals in the algebra having finite spectrum. We use Fredholm index theory to modify and simplify this construction to obtain such examples that are not limits of any ``nice'' types of elements.

**1.**L. G. Brown, R. G. Douglas, and P. A. 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) Springer, Berlin, 1973, pp. 58–128. Lecture Notes in Math., Vol. 345. MR**0380478****2.**John B. Conway,*The theory of subnormal operators*, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR**1112128****3.**K. F. Clancy,*Seminormal operators*, Lecture Notes in Math., vol. 742, Springer-Verlag, New York, 1979.**4.**Don Hadwin,*Strongly quasidiagonal 𝐶*-algebras*, J. Operator Theory**18**(1987), no. 1, 3–18. With an appendix by Jonathan Rosenberg. MR**912809****5.**Terry A. Loring,*Normal elements of 𝐶*-algebras of real rank zero without finite-spectrum approximants*, J. London Math. Soc. (2)**51**(1995), no. 2, 353–364. MR**1325578**, 10.1112/jlms/51.2.353**6.**Terry A. Loring and Jack Spielberg,*Approximation of normal elements in the multiplier algebra of an AF 𝐶*-algebra*, Proc. Amer. Math. Soc.**121**(1994), no. 4, 1173–1175. MR**1211584**, 10.1090/S0002-9939-1994-1211584-5

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46L80,
47A58,
46L05,
47C15

Retrieve articles in all journals with MSC (1991): 46L80, 47A58, 46L05, 47C15

Additional Information

**Don Hadwin**

Affiliation:
Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824

Email:
don@math.unh.edu

**Terry A. Loring**

Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131

Email:
loring@deepthought.unm.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-97-03734-9

Received by editor(s):
July 3, 1995

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society