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On the algebra range of an operator on a Hilbert $C^*$-module over compact operators


Author: Rajna Rajic
Journal: Proc. Amer. Math. Soc. 131 (2003), 3043-3051
MSC (2000): Primary 47A12, 46L08
Published electronically: May 5, 2003
MathSciNet review: 1993211
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a Hilbert $C^*$-module over the $C^*$-algebra $K(H)$ of all compact operators on a complex Hilbert space $H$. Given an orthogonal projection $p \in K(H)$, we describe the set $ V^n(A) = \{\langle Ax,x\rangle \,\,:\,\,x\in X,\,\,\langle x,x \rangle =p\} $ for an arbitrary adjointable operator $A\in B(X)$. The relationship between the set $ V^n(A)$ and the matricial range of $A$ is established.


References [Enhancements On Off] (What's this?)

  • 1. William B. Arveson, Subalgebras of 𝐶*-algebras, Acta Math. 123 (1969), 141–224. MR 0253059
  • 2. William Arveson, Subalgebras of 𝐶*-algebras. II, Acta Math. 128 (1972), no. 3-4, 271–308. MR 0394232
  • 3. D. Bakic and B. Guljas, Hilbert $C^*$-modules over $C^*$-algebras of compact operators, Acta Sci. Math. (Szeged) 68 (2002), 249-269.
  • 4. F. F. Bonsall and J. Duncan, Numerical ranges. II, Cambridge University Press, New York-London, 1973. London Mathematical Society Lecture Notes Series, No. 10. MR 0442682
  • 5. John Bunce and Norberto Salinas, Completely positive maps on 𝐶*-algebras and the left matricial spectra of an operator, Duke Math. J. 43 (1976), no. 4, 747–774. MR 0430793
  • 6. E. C. Lance, Hilbert 𝐶*-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694
  • 7. 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
  • 8. J. R. Ringrose, Compact non-selfadjoint operators, Van Nostrand, Princeton, 1971.
  • 9. R. R. Smith and J. D. Ward, Matrix ranges for Hilbert space operators, Amer. J. Math. 102 (1980), no. 6, 1031–1081. MR 595006, 10.2307/2374180

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Additional Information

Rajna Rajic
Affiliation: Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
Email: rajna.rajic@zg.hinet.hr

DOI: https://doi.org/10.1090/S0002-9939-03-07130-2
Keywords: $C^*$-algebra, Hilbert $C^*$-module, adjointable operator, matricial range of an operator
Received by editor(s): June 20, 2001
Received by editor(s) in revised form: January 22, 2002
Published electronically: May 5, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society