## Extreme points in triangular UHF algebras

HTML articles powered by AMS MathViewer

- by Timothy D. Hudson, Elias G. Katsoulis and David R. Larson PDF
- Trans. Amer. Math. Soc.
**349**(1997), 3391-3400 Request permission

## Abstract:

We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points.## References

- M. Anoussis and E.G. Katsoulis,
*Finite rank operators and the geometric structure of nest algebras*, preprint, 1995. - —,
*A non-selfadjoint Russo-Dye theorem*, Math. Ann.**304**(1996), 685–699. - M. Anoussis and E. G. Katsoulis,
*Compact operators and the geometric structure of $C^*$-algebras*, Proc. Amer. Math. Soc.**124**(1996), no. 7, 2115–2122. MR**1322911**, DOI 10.1090/S0002-9939-96-03285-6 - Jonathan Arazy and Baruch Solel,
*Isometries of nonselfadjoint operator algebras*, J. Funct. Anal.**90**(1990), no. 2, 284–305. MR**1052336**, DOI 10.1016/0022-1236(90)90085-Y - J. A. Cima and James Thomson,
*On strong extreme points in $H^{p}$*, Duke Math. J.**40**(1973), 529–532. MR**318894** - Kenneth R. Davidson, Timothy G. Feeman, and Allen L. Shields,
*Extreme points in quotients of operator algebras*, Topics in operator theory, Oper. Theory Adv. Appl., vol. 32, Birkhäuser, Basel, 1988, pp. 67–91. MR**951957**, DOI 10.1007/978-3-0348-5475-7_{7} - Mahlon M. Day,
*Normed linear spaces*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR**0344849** - Allan P. Donsig,
*Semisimple triangular AF algebras*, J. Funct. Anal.**111**(1993), no. 2, 323–349. MR**1203457**, DOI 10.1006/jfan.1993.1016 - A.P. Donsig and T.D. Hudson,
*The lattice of ideals of a triangular AF algebra*, J. Funct. Anal.**138**(1996), 1–39. - Alan Hopenwasser and Justin R. Peters,
*Full nest algebras*, Illinois J. Math.**38**(1994), no. 3, 501–520. MR**1269701** - T. D. Hudson,
*Ideals in triangular AF algebras*, Proc. London Math. Soc. (3)**69**(1994), no. 2, 345–376. MR**1281969**, DOI 10.1112/plms/s3-69.2.345 - Henryk Hudzik, Wiesław Kurc, and Marek Wisła,
*Strongly extreme points in Orlicz function spaces*, J. Math. Anal. Appl.**189**(1995), no. 3, 651–670. MR**1312545**, DOI 10.1006/jmaa.1995.1043 - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. II*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR**859186**, DOI 10.1016/S0079-8169(08)60611-X - Robert McGuigan,
*Strongly extreme points in Banach spaces*, Manuscripta Math.**5**(1971), 113–122. MR**308743**, DOI 10.1007/BF01325021 - R. L. Moore and T. T. Trent,
*Extreme points of certain operator algebras*, Indiana Univ. Math. J.**36**(1987), no. 3, 645–650. MR**905616**, DOI 10.1512/iumj.1987.36.36036 - R. L. Moore and T. T. Trent,
*Isometries of nest algebras*, J. Funct. Anal.**86**(1989), no. 1, 180–209. MR**1013938**, DOI 10.1016/0022-1236(89)90069-4 - Paul S. Muhly, Chao Xin Qiu, and Baruch Solel,
*On isometries of operator algebras*, J. Funct. Anal.**119**(1994), no. 1, 138–170. MR**1255276**, DOI 10.1006/jfan.1994.1006 - 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 - Stephen C. Power,
*Limit algebras: an introduction to subalgebras of $C^*$-algebras*, Pitman Research Notes in Mathematics Series, vol. 278, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR**1204657** - —,
*On the Banach space isomorphism type of AF $C^*$-algebras and their triangular subalgebras*, Israel J. Math.**94**(1996), 93–109. - A. Guyan Robertson,
*A note on the unit ball in $C^{\ast }$-algebras*, Bull. London Math. Soc.**6**(1974), 333–335. MR**355621**, DOI 10.1112/blms/6.3.333

## Additional Information

**Timothy D. Hudson**- Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858-4353
- Email: tdh@.math.ecu.edu
**Elias G. Katsoulis**- Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858-4353
- MR Author ID: 99165
- Email: makatsov@ecuvm.cis.ecu.edu
**David R. Larson**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- MR Author ID: 110365
- Email: larson@math.tamu.edu
- Received by editor(s): January 11, 1996
- Received by editor(s) in revised form: March 28, 1996
- Additional Notes: The first author’s research was partially supported by NSF grant #DMS-9500566 and the Linear Analysis and Probability Workshop at Texas A&M University.

The second author’s research was partially supported by a YI grant from the Linear Analysis and Probability Workshop at Texas A&M University and a grant from East Carolina University.

The third author’s research was partially supported by NSF grant #DMS-9401544. - © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**349**(1997), 3391-3400 - MSC (1991): Primary 47D25, 46K50, 46B20
- DOI: https://doi.org/10.1090/S0002-9947-97-01882-5
- MathSciNet review: 1407493