Illinois Journal of Mathematics

Finite dimensional point derivations for graph algebras

Benton L. Duncan

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Abstract

This paper focuses on finite dimensional point derivations for the non-self-adjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor algebra of a directed graph. We determine when such a derivation is inner, and describe situations that give rise to noninner derivations. We also analyze the situation when the derivation corresponds to a multiplicative linear functional.

Article information

Source
Illinois J. Math., Volume 52, Number 2 (2008), 419-435.

Dates
First available in Project Euclid: 23 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1248355342

Digital Object Identifier
doi:10.1215/ijm/1248355342

Mathematical Reviews number (MathSciNet)
MR2524644

Zentralblatt MATH identifier
1191.47085

Subjects
Primary: 47L40: Limit algebras, subalgebras of $C^*$-algebras
Secondary: 47L55: Representations of (nonselfadjoint) operator algebras 47L75: Other nonselfadjoint operator algebras 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]

Citation

Duncan, Benton L. Finite dimensional point derivations for graph algebras. Illinois J. Math. 52 (2008), no. 2, 419--435. doi:10.1215/ijm/1248355342. https://projecteuclid.org/euclid.ijm/1248355342


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References

  • M. Alaimia, Automorphisms of some Banach algebras of analytic functions, Linear Algebra Appl. 298 (1999), 87–97.
  • F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973.
  • A. Browder, Introduction to Function Algebras, W. A. Benjamin, New York, 1969.
  • F. Gilfeather, Derivations on certain CSL algebras, J. Operator Theory 11 (1984), 145–156.
  • K. Davidson and E. Katsoulis, Nest representations of directed graph algebras, Proc. London Math. Soc. 92 (2006), 762–790.
  • K. Davidson and D. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), 275–303.
  • K. Davidson and D. Pitts, Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras, Integral Equations Operator Theory 31 (1998), 321–337.
  • B. Duncan, Noncommutative point derivations for matrix function algebras, to appear in Linear Algebra Appl.
  • B. Duncan, Automorphisms of nonselfadjoint directed graph operator algebras, to appear in J. Austral. Math. Soc.
  • F. Jaeck and S. Power, Hyper-reflexivity of free-semigroupoid algebras, Proc. Amer. Math. Soc. 134 (2006), 2027–2035.
  • B. Johnson, Cohomology in Banach Algebras, Mem. Amer. Math. Soc., vol. 127, Amer. Math. Soc, Providence, RI, 1972.
  • M. Jury and D. Kribs, Partially isometric dilations of noncommuting $N$-tuples of operators, Proc. Amer. Math. Soc. 133 (2005), 213–222.
  • M. Jury and D. Kribs, Ideal structure in free semigroupoid algebras from directed graphs, J. Operator Theory 53 (2005), 273–302.
  • E. Katsoulis and D. Kribs, Isomorphisms of algebras associated with directed graphs, Math. Ann. 330 (2004), 709–728.
  • D. Kribs and S. Power, Free semigroupoid algebras, J. Ramanujan Math. Soc. 19 (2004), 75–114.
  • D. Kribs and S. Power, Partly free algebras from directed graphs, Current Trends in Operator Theory and Its Applications, Birkhäuser, Basel, 2004, pp. 373–385.
  • P. Muhly and B. Solel, Tensor algebras over $C^*$-correspondences: representations, dilations, and $C^*$-envelopes, J. Funct. Anal. 158(2) (1998), 389–457.
  • G. Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124(7) (1996), 2137–2148.