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Isometries between matrix algebras

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

Wai-Shun Cheung ,
Chi-Kwong Li  and
Yiu-Tung Poon
Wai-Shun Cheung
Affiliation:
Center of Linear Algebra and Combinatorics, University of Lisbon, Lisbon, Portugal
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795, USA e-mail: ckli@math.wm.edu
Yiu-Tung Poon
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011USA e-mail: ytpoon@iastate.edu
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Abstract

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As an attempt to understand linear isometries between C*-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m × m complex matrices. If k ≥ n and φ: Mn → Mk has the form X ↦ U[X ⊕ f(X)] V or X ↦ U[X1 ⊕ f(X)]V for some unitary U, V ∈ Mk and contractive linear map f: Mn → Mk, then ║φ(X)║ = ║X║ for all X ∈ Mn. We prove that the converse is true if k ≤ 2n - 1, and the converse may fail if k ≥ 2n. Related results and questions involving positive linear maps and the numerical range are discussed.

Keywords

isometrymatriceslinear maps

MSC classification

Secondary: 15A04: Linear transformations, semilinear transformations 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 1 , August 2004 , pp. 1 - 16
Copyright
Copyright © Australian Mathematical Society 2004

References

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