Abstract
We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras. These are the weak operator topology closed algebras generated by the left regular representations of semigroupoids associated with finite or countable directed graphs. We expand our analysis of partly free algebras from previous work and obtain a graph-theoretic characterization of when a free semigroupoid algebra with countable graph is partly free. This analysis carries over to norm closed quiver algebras. We also discuss new examples for the countable graph case.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. Arias, Multipliers and representations of noncommutative disc algebras, Houston J. Math., 25 (1999), 99–120.
A. Arias, G. Popescu, Factorization and reflexivity on Fock spaces, Int. Equat. Oper. Th. 23 (1995), 268–286.
W. Arveson, Interpolation problems in nest algebras, J. Func. Anal. 20 (1975), 208–233.
T. Bates, J. Hong, I. Raeburn, W. Szymenski, The ideal structure of the C* -algebras of infinite graphs, e-print arxiv math.OA/0109142, preprint, 2001.
H. Bercovici, Hyper-reflexivity and the factorization of linear functionals, J. Func. Anal. 158 (1998), 242–252.
A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239–255.
E. Christensen, Perturbations of operator algebras II, Indiana U. Math. J. 26 (1977), 891–904.
K.R. Davidson, The distance to the analytic Toeplitz operators, Illinois J. Math. 31 (1987), 265–273.
K.R. Davidson, E. Katsoulis, D.R. Pitts, The structure of free semigroup algebras, J. reine angew. Math. 533 (2001), 99–125.
K.R. Davidson, D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semi-group algebras, Proc. London Math. Soc. 78 (1999), 401–430.
K.R. Davidson, D.R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275–303.
K.R. Davidson, Nest Algebras, Longman Scientific & Technical, London, 1988.
L.M. De Alba, J. Peters, Classification of semicrossed products of finite-dimensional C* -algebras, Proc. Amer. Math. Soc. 95 (1985), 557–564.
R. Douglas, Banach algebra techniques in operator theory, Springer-Verlag, New York, 1998.
M. Ephrem, Characterizing liminal and type I graph C* -algebras, arXiv:math.OA/0211241, preprint, 2003.
K. Hoffman, Banach spaces of analytic functions, Dover Publications Inc., New York, 1988.
F. Jaeck, S.C. Power, The semigroupoid algebras of finite graphs are hyper-reflexive, in preparation, 2003.
M.T. Jury, D.W. Kribs, Ideal structure in free semigroupoid algebras from directed graphs, preprint, 2003.
D.W. Kribs, S.C. Power, Free semigroupoid algebras, preprint, 2002.
D.W. Kribs, Factoring in non-commutative analytic Toeplitz algebras, J. Operator Theory 45 (2001), 175–193.
A. Kumjian, D. Pask, I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math 184 (1998), 161–174.
A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, Groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505–541.
P.S. Muhly, A finite-dimensional introduction to operator algebra, A. Katavolos (ed.), Operator Algebras and Applications, 313–354, Kluwer Academic Publishers, 1997.
P.S. Muhly, B. Solel, Tensor algebras, induced representations, and the Wold decomposition, Can. J. Math. 51 (4), 1999, 850–880.
G. Popescu, Multi-analytic operators and some factorization theorems, Indiana Univ. Math. J. 38 (1989), 693–710.
G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46.
G. Popescu, Noncommuting disc algebras and their representations, Proc. Amer. Math. Soc. 124 (1996), 2137–2148.
S.C. Power, Approximately finitely acting operator algebras, J. Func. Anal. 189 (2002), 409–469.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Basel AG
About this paper
Cite this paper
Kribs, D.W., Power, S.C. (2004). Partly Free Algebras From Directed Graphs. In: Ball, J.A., Helton, J.W., Klaus, M., Rodman, L. (eds) Current Trends in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 149. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7881-4_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7881-4_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9608-5
Online ISBN: 978-3-0348-7881-4
eBook Packages: Springer Book Archive