Hyper-reflexivity of free semigroupoid algebras

Authors:
Frédéric Jaëck and Stephen C. Power

Journal:
Proc. Amer. Math. Soc. **134** (2006), 2027-2035

MSC (2000):
Primary 47L75

Published electronically:
December 19, 2005

MathSciNet review:
2215772

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra is generated by a family of partial isometries, and initial projections, which act on a generalized Fock space spawned by the directed graph . We show that if the graph is finite, then is hyper-reflexive.

**1.**Alvaro Arias and Gelu Popescu,*Factorization and reflexivity on Fock spaces*, Integral Equations Operator Theory**23**(1995), no. 3, 268–286. MR**1356335**, 10.1007/BF01198485**2.**William Arveson,*Ten lectures on operator algebras*, CBMS Regional Conference Series in Mathematics, vol. 55, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1984. MR**762819****3.**Hari Bercovici,*Hyper-reflexivity and the factorization of linear functionals*, J. Funct. Anal.**158**(1998), no. 1, 242–252. MR**1641578**, 10.1006/jfan.1998.3288**4.**Hari Bercovici, Ciprian Foias, and Carl Pearcy,*Dual algebras with applications to invariant subspaces and dilation theory*, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR**787041****5.**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****6.**Kenneth R. Davidson,*The distance to the analytic Toeplitz operators*, Illinois J. Math.**31**(1987), no. 2, 265–273. MR**882114****7.**Kenneth R. Davidson, Elias Katsoulis, and David R. Pitts,*The structure of free semigroup algebras*, J. Reine Angew. Math.**533**(2001), 99–125. MR**1823866**, 10.1515/crll.2001.028**8.**Kenneth R. Davidson and David R. Pitts,*Invariant subspaces and hyper-reflexivity for free semigroup algebras*, Proc. London Math. Soc. (3)**78**(1999), no. 2, 401–430. MR**1665248**, 10.1112/S002461159900180X**9.**Kenneth R. Davidson and David R. Pitts,*The algebraic structure of non-commutative analytic Toeplitz algebras*, Math. Ann.**311**(1998), no. 2, 275–303. MR**1625750**, 10.1007/s002080050188**10.**David W. Kribs and Stephen C. Power,*Free semigroupoid algebras*, J. Ramanujan Math. Soc.**19**(2004), no. 2, 117–159. MR**2076898****11.**David W. Kribs and Stephen C. Power,*Partly free algebras from directed graphs*, Current trends in operator theory and its applications, Oper. Theory Adv. Appl., vol. 149, Birkhäuser, Basel, 2004, pp. 373–385. MR**2063759****12.**Gelu Popescu,*Non-commutative disc algebras and their representations*, Proc. Amer. Math. Soc.**124**(1996), no. 7, 2137–2148. MR**1343719**, 10.1090/S0002-9939-96-03514-9**13.**Gelu Popescu,*A generalization of Beurling’s theorem and a class of reflexive algebras*, J. Operator Theory**41**(1999), no. 2, 391–420. MR**1681580**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47L75

Retrieve articles in all journals with MSC (2000): 47L75

Additional Information

**Frédéric Jaëck**

Affiliation:
University of Bordeaux I, LaBAG 351, cours de la Liberation, F-33405 Talence, Cedex, France

Email:
jaeck@math.u-bordeaux1.fr

**Stephen C. Power**

Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster, Lancashire LA1, England

Email:
s.power@lancaster.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-05-08209-2

Keywords:
Free semigroupoid,
reflexive algebra,
directed graph,
hyper-reflexive

Received by editor(s):
February 10, 2005

Published electronically:
December 19, 2005

Additional Notes:
This work is part of the research program of the network “Analysis and Operators" supported by the European Community’s Potential Program under HPRN-CT-2000-00116 (Analysis and operators).

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.