J. Aust. Math. Soc.
77 (2004), 116

Isometries between matrix algebras

WaiShun 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


ChiKwong Li
Department of Mathematics
College of William and Mary
P.O. Box 8795
Williamsburg
Virginia 231878795
USA
ckli@math.wm.edu



YiuTung Poon
Department of Mathematics
Iowa State University
Ames
Iowa 50011
USA
ytpoon@iastate.edu



Abstract

As an attempt to understand linear isometries
between algebras without the surjectivity assumption, we
study linear isometries between matrix algebras.
Denote by the algebra of
complex matrices.
If and has the form
or
for some unitary and contractive linear map
, then
for all
. We prove that the converse is true if
, and the converse may fail if
. Related results and questions involving
positive linear maps and the numerical range are
discussed.

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