J. Aust. Math. Soc.
81 (2006), 165184

Mappings on matrices: invariance of functional values of matrix products

JorTing Chan
Department of Mathematics
The University of Hong Kong
Pokfulam Road
Hong Kong
jtchan@hku.hk


ChiKwong Li
Department of Mathematics
College of William and Mary
Williamsburg, VA 231878795
USA
ckli@math.wm.edu



NungSing Sze
Department of Mathematics
The University of Hong Kong
Pokfulam Road
Hong Kong
and
Department of Mathematics
University of Connecticut
Storrs, CT 062693009
USA
sze@math.uconn.edu



Abstract

Let
be the algebra of all matrices over a field
, where
. Let
be a subset of
containing all rank one matrices. We study
mappings
such that
for various families of functions
including all the unitary similarity invariant
functions on real or complex matrices. Very
often, these mappings have the form
for all
for some invertible , field monomorphism
of
, and an
valued mapping
defined on
. For real matrices,
is often the identity map; for complex matrices,
is often the identity map or the conjugation
map:
. A key idea in our study is reducing the problem
to the special case when
is defined by
, if
, and
otherwise. In such a case, one needs to
characterize
such that
if and only if
. We show that such a map has the standard form
described above on rank one matrices in
.

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