J. Aust. Math. Soc.  77 (2004), 1-16
Isometries between matrix algebras

Wai-Shun Cheung
  Center of Linear Algebra and Combinatorics
  University of Lisbon
  Lisbon
  Portugal
  Current address:
  Department of Mathematics
  University of Hong Kong
  Hong Kong
  a9000068@yahoo.com
Chi-Kwong Li
  Department of Mathematics
  College of William and Mary
  P.O. Box 8795
  Williamsburg
  Virginia 23187-8795
  USA
  ckli@math.wm.edu
and
Yiu-Tung Poon
  Department of Mathematics
  Iowa State University
  Ames
  Iowa 50011
  USA
  ytpoon@iastate.edu


Abstract
As an attempt to understand linear isometries between $C^*$-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by $M_m$ the algebra of $m\times m$ complex matrices. If $k \ge n$ and $\phi: M_n \rightarrow M_k$ has the form $X \mapsto U[X \oplus f(X)]V$ or $X \mapsto U[X^t \oplus f(X)]V$ for some unitary $U, V \in M_k$ and contractive linear map $f:M_n \to M_{k}$, then $\|\phi(X)\| = \|X\|$ for all $X \in M_n$. We prove that the converse is true if $k \le 2n-1$, and the converse may fail if $k \ge 2n$. Related results and questions involving positive linear maps and the numerical range are discussed.
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