The norm of a symmetric elementary operator
HTML articles powered by AMS MathViewer
- by Bojan Magajna PDF
- Proc. Amer. Math. Soc. 132 (2004), 1747-1754 Request permission
Abstract:
The norm of the operator $x\mapsto a^*xb+b^*xa$ on $A= {\mathrm B}(\mathcal {H})$ (or on any prime C$^*$-algebra $A$) is computed for all $a,b\in A$ and is shown to be equal to the completely bounded norm.References
- Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
- Lawrence A. Fialkow, Structural properties of elementary operators, Elementary operators and applications (Blaubeuren, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 55–113. MR 1183937
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
- Bojan Magajna, The Haagerup norm on the tensor product of operator modules, J. Funct. Anal. 129 (1995), no. 2, 325–348. MR 1327181, DOI 10.1006/jfan.1995.1053
- B. Magajna and A. Turnšek, On the norm of symmetrised two-sided multiplications, Bull. Austral. Math. Soc. 67 (2003), 27–38.
- Martin Mathieu, Properties of the product of two derivations of a $C^*$-algebra, Canad. Math. Bull. 32 (1989), no. 4, 490–497. MR 1019418, DOI 10.4153/CMB-1989-072-4
- Martin Mathieu, The norm problem for elementary operators, Recent progress in functional analysis (Valencia, 2000) North-Holland Math. Stud., vol. 189, North-Holland, Amsterdam, 2001, pp. 363–368. MR 1861772, DOI 10.1016/S0304-0208(01)80061-X
- Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
- R. R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991), no. 1, 156–175. MR 1138841, DOI 10.1016/0022-1236(91)90139-V
- L. L. Stachó and B. Zalar, On the norm of Jordan elementary operators in standard operator algebras, Publ. Math. Debrecen 49 (1996), no. 1-2, 127–134. MR 1416312
- L. L. Stachó and B. Zalar, Uniform primeness of the Jordan algebra of symmetric operators, Proc. Amer. Math. Soc. 126 (1998), no. 8, 2241–2247. MR 1487342, DOI 10.1090/S0002-9939-98-04769-8
- Joseph G. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737–747. MR 265952
- Richard M. Timoney, A note on positivity of elementary operators, Bull. London Math. Soc. 32 (2000), no. 2, 229–234. MR 1734184, DOI 10.1112/S0024609399006815
Additional Information
- Bojan Magajna
- Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia
- Email: Bojan.Magajna@fmf.uni-lj.si
- Received by editor(s): July 19, 2002
- Received by editor(s) in revised form: February 7, 2003
- Published electronically: October 8, 2003
- Additional Notes: Supported by the Ministry of Science and Education of Slovenia
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1747-1754
- MSC (2000): Primary 47B47
- DOI: https://doi.org/10.1090/S0002-9939-03-07248-4
- MathSciNet review: 2051136