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Mathematical Methods in Systems, Optimization, and Control pp 279–293Cite as

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Positive Completion Problems Over C*-algebras

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 222))

Abstract

We study how positive completion problems over matrices with complex entries generalize to the setting of matrices with C*-algebra entries. In particular, it is observed that some C*-algebras have the Toeplitz banded completion property but fail to have the completion property for the complete graph with one undirected edge missing. Positive completions in the framework of multi-level Toeplitz matrices are also studied. Many open problems are formulated.

Mathematics Subject Classification. 46L05 (47A57).

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References

  1. Mihály Bakonyi and Geir Nævdal. The finite subsets of Z2 having the extensión property. J. London Math. Soc. (2), 62(3):904-916, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  2. Mihály Bakonyi and Hugo J. Woerdeman. Matrix completions, moments, and sums of Hermitian squares. Princeton University Press, Princeton, NJ, 2011

    Google Scholar 

  3. Mihály Bakonyi and Dan Timotin. The central completion of a positive block operator matrix. Operator theory, structured matrices, and dilations, Theta Ser. Adv. Math., 7:69–83, 2007; Theta, Bucharest.

    Google Scholar 

  4. Kenneth R. Davidson. Lifting positive elements in C*-algebras. Integral Equations Operator Theory, 14(2):183-191, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  5. Jean-Pierre Gabardo. Trigonometric moment problems for arbitrary finite subsets of Zn. Trans. Amer. Math. Soc., 350(11):4473–4498, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  6. Jeffrey S. Geronimo and Hugo J. Woerdeman. Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables. Ann. of Math. (2), 160(3):839–906, 2004.

    Article  MathSciNet  Google Scholar 

  7. Jeffrey S. Geronimo and Hugo J. Woerdeman. The operator-valued autoregressive fil- ter problem and the suboptimal Nehari problem in two variables. Integral Equations Operator Theory, 53(3):343-361, 2005.

    MATH  Google Scholar 

  8. Martin Charles Golumbic. Algorithmic graph theory and perfect graphs, volume 57 of Annals of Discrete Mathematics. Elsevier Science B.V., Amsterdam, second edition, 2004.

    Google Scholar 

  9. Robert Grone, Charles R. Johnson, Eduardo M. de Sá, and Henry Wolkowicz. Positive definite completions of partial Hermitian matrices. Linear Algebra Appl., 58:109124, 1984.

    Google Scholar 

  10. John L. Kelly. General Topology. D. Van Nostrand Company, Princeton, New Jersey, 1957.

    Google Scholar 

  11. Jan van Mill, private communication.

    Google Scholar 

  12. Gerard J. Murphy. C*-algebras and operator theory. Academic Press Inc., Boston, MA, 1990.

    Google Scholar 

  13. Vern I. Paulsen and Leiba Rodman. Positive completions of matrices over - algebras. J. Operator Theory, 25(2):237–253, 1991.

    MathSciNet  MATH  Google Scholar 

  14. L.A. Sahnovič. Effective construction of noncontinuable Hermite-positive functions of several variables. Funktsional. Anal. i Prilozhen., 14(4):55–60, 96, 1980.

    MathSciNet  Google Scholar 

  15. L.A. Sakhnovich. Interpolation theory and its applications, volume 428 of Mathemat- ics and its Applications. Kluwer Academic, Dordrecht, 1997.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of William and Mary, Williamsburg, VA, USA

    L. Rodman

  2. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    H. J. Woerdeman

Authors
  1. L. Rodman
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  2. H. J. Woerdeman
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Corresponding author

Correspondence to L. Rodman .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Harry Dym

  2. , Department of Mechanical and, University of California San Diego, Gilman Drive 9500, La Jolla, 92093-0411, California, USA

    Mauricio C. de Oliveira

  3. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

Additional information

Dedicated to J.W. Helton on the occasion of his 65th birthday

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Rodman, L., Woerdeman, H.J. (2012). Positive Completion Problems Over C*-algebras. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_20

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