## Compact operators and nest representations of limit algebras

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- by Elias Katsoulis and Justin R. Peters PDF
- Trans. Amer. Math. Soc.
**359**(2007), 2721-2739 Request permission

## Abstract:

In this paper we study the nest representations $\rho : \mathcal {A} \longrightarrow \operatorname {Alg} \mathcal {N}$ of a strongly maximal TAF algebra $\mathcal {A}$, whose ranges contain non-zero compact operators. We introduce a particular class of such representations, the essential nest representations, and we show that their kernels coincide with the completely meet irreducible ideals. From this we deduce that there exist enough contractive nest representations, with non-zero compact operators in their range, to separate the points in $\mathcal {A}$. Using nest representation theory, we also give a coordinate-free description of the fundamental groupoid for strongly maximal TAF algebras. For an arbitrary nest representation $\rho : \mathcal {A} \longrightarrow \operatorname {Alg} \mathcal {N}$, we show that the presence of non-zero compact operators in the range of $\rho$ implies that $\mathcal {N}$ is similar to a completely atomic nest. If, in addition, $\rho (\mathcal {A} )$ is closed, then every compact operator in $\rho (\mathcal {A} )$ can be approximated by sums of rank one operators $\rho (\mathcal {A} )$. In the case of $\mathbb {N}$-ordered nest representations, we show that $\rho ( \mathcal {A})$ contains finite rank operators iff $\ker \rho$ fails to be a prime ideal.## References

- Kenneth R. Davidson,
*Nest algebras*, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR**972978** - Kenneth R. Davidson,
*$C^*$-algebras by example*, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR**1402012**, DOI 10.1090/fim/006 - Kenneth R. Davidson and Elias Katsoulis,
*Primitive limit algebras and $C^*$-envelopes*, Adv. Math.**170**(2002), no. 2, 181–205. MR**1932328**, DOI 10.1006/aima.2001.2068 - Kenneth R. Davidson, Elias Katsoulis, and Justin Peters,
*Meet-irreducible ideals and representations of limit algebras*, J. Funct. Anal.**200**(2003), no. 1, 23–30. MR**1974086**, DOI 10.1016/S0022-1236(02)00058-7 - Allan P. Donsig, Alan Hopenwasser, Timothy D. Hudson, Michael P. Lamoureux, and Baruch Solel,
*Meet irreducible ideals in direct limit algebras*, Math. Scand.**87**(2000), no. 1, 27–63. MR**1776964**, DOI 10.7146/math.scand.a-14298 - Allan P. Donsig and Timothy D. Hudson,
*The lattice of ideals of a triangular AF algebra*, J. Funct. Anal.**138**(1996), no. 1, 1–39. MR**1391629**, DOI 10.1006/jfan.1996.0055 - A. P. Donsig, T. D. Hudson, and E. G. Katsoulis,
*Algebraic isomorphisms of limit algebras*, Trans. Amer. Math. Soc.**353**(2001), no. 3, 1169–1182. MR**1804417**, DOI 10.1090/S0002-9947-00-02714-8 - Allan P. Donsig, David R. Pitts, and S. C. Power,
*Algebraic isomorphisms and spectra of triangular limit algebras*, Indiana Univ. Math. J.**50**(2001), no. 3, 1131–1147. MR**1871350**, DOI 10.1512/iumj.2001.50.2113 - J. A. Erdos,
*Operators of finite rank in nest algebras*, J. London Math. Soc.**43**(1968), 391–397. MR**230156**, DOI 10.1112/jlms/s1-43.1.391 - J. A. Erdos and S. C. Power,
*Weakly closed ideals of nest algebras*, J. Operator Theory**7**(1982), no. 2, 219–235. MR**658610** - James Glimm,
*Type I $C^{\ast }$-algebras*, Ann. of Math. (2)**73**(1961), 572–612. MR**124756**, DOI 10.2307/1970319 - Alan Hopenwasser, Justin R. Peters, and Stephen C. Power,
*Nest representations of TAF algebras*, Canad. J. Math.**52**(2000), no. 6, 1221–1234. MR**1794303**, DOI 10.4153/CJM-2000-051-7 - Elias Katsoulis and David W. Kribs,
*Isomorphisms of algebras associated with directed graphs*, Math. Ann.**330**(2004), no. 4, 709–728. MR**2102309**, DOI 10.1007/s00208-004-0566-6 - E. G. Katsoulis and R. L. Moore,
*On compact operators in certain reflexive operator algebras*, J. Operator Theory**25**(1991), no. 1, 177–182. MR**1191259** - David W. Kribs and Stephen C. Power,
*Free semigroupoid algebras*, J. Ramanujan Math. Soc.**19**(2004), no. 2, 117–159. MR**2076898** - Michael P. Lamoureux,
*Nest representations and dynamical systems*, J. Funct. Anal.**114**(1993), no. 2, 467–492. MR**1223711**, DOI 10.1006/jfan.1993.1075 - Michael P. Lamoureux,
*Ideals in some continuous nonselfadjoint crossed product algebras*, J. Funct. Anal.**142**(1996), no. 1, 211–248. MR**1419421**, DOI 10.1006/jfan.1996.0148 - Michael P. Lamoureux,
*The topology of ideals in some triangular AF algebras*, J. Operator Theory**37**(1997), no. 1, 91–109. MR**1438202** - David R. Larson and Baruch Solel,
*Structured triangular limit algebras*, Proc. London Math. Soc. (3)**75**(1997), no. 1, 177–193. MR**1444318**, DOI 10.1112/S0024611597000312 - Paul S. Muhly,
*A finite-dimensional introduction to operator algebra*, Operator algebras and applications (Samos, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 495, Kluwer Acad. Publ., Dordrecht, 1997, pp. 313–354. MR**1462686** - Paul S. Muhly and Baruch Solel,
*Subalgebras of groupoid $C^*$-algebras*, J. Reine Angew. Math.**402**(1989), 41–75. MR**1022793**, DOI 10.1515/crll.1989.402.41 - John L. Orr and Justin R. Peters,
*Some representations of TAF algebras*, Pacific J. Math.**167**(1995), no. 1, 129–161. MR**1318167**, DOI 10.2140/pjm.1995.167.129 - Vern I. Paulsen,
*Completely bounded maps and dilations*, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR**868472** - Gert K. Pedersen,
*$C^{\ast }$-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006** - J. R. Peters, Y. T. Poon, and B. H. Wagner,
*Triangular AF algebras*, J. Operator Theory**23**(1990), no. 1, 81–114. MR**1054818** - S. C. Power,
*Classification of tensor products of triangular operator algebras*, Proc. London Math. Soc. (3)**61**(1990), no. 3, 571–614. MR**1069516**, DOI 10.1112/plms/s3-61.3.571 - J. R. Ringrose,
*On some algebras of operators*, Proc. London Math. Soc. (3)**15**(1965), 61–83. MR**171174**, DOI 10.1112/plms/s3-15.1.61 - Şerban Strătilă and Dan Voiculescu,
*Representations of AF-algebras and of the group $U(\infty )$*, Lecture Notes in Mathematics, Vol. 486, Springer-Verlag, Berlin-New York, 1975. MR**0458188**, DOI 10.1007/BFb0082276 - Michael Thelwall,
*Dilation theory for subalgebras of AF algebras*, J. Operator Theory**25**(1991), no. 2, 275–282. MR**1203033**

## Additional Information

**Elias Katsoulis**- Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
- MR Author ID: 99165
- Email: katsoulise@ecu.edu
**Justin R. Peters**- Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
- Email: peters@iastate.edu
- Received by editor(s): April 15, 2004
- Received by editor(s) in revised form: March 27, 2005
- Published electronically: January 4, 2007
- Additional Notes: The first author’s research was partially supported by a grant from ECU
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 2721-2739 - MSC (2000): Primary 47L80
- DOI: https://doi.org/10.1090/S0002-9947-07-04071-8
- MathSciNet review: 2286053