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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

[1]Cheung, W. S. and Li, C. K., ‘Linear maps transforming the unitary group’, Canad. Math. Bulletin 46 (2003), 54–58.CrossRefGoogle Scholar
[2]Chu, C.-H. and Wong, N.-C., ‘Isometries between C*-algebras’, Revista Matematica Iberoamericana (to appear).Google Scholar
[3]Helton, J. W. and Rodman, L., ‘Signature preserving linear maps on Hermitian matrices’, Linear and Multilinear Algebra 17 (1985), 29–37.CrossRefGoogle Scholar
[4]Horn, R. and Johnson, C., Matrix analysis (Cambridge University Press, Cambridge, 1985).CrossRefGoogle Scholar
[5]Horn, R. and Johnson, C., Topics in matrix analysis (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[6]Kadison, R. V., ‘Isometries of operator algebras’, Ann. of Math. 54 (1951), 325–338.Google Scholar
[7]Thompson, R. C., ‘Principal submatrices. IX. Interlacing inequalities for singular values of submatrices’, Linear Algebra Appl. 5 (1972), 1–12.CrossRefGoogle Scholar