Abstract
On the ring of all n × n complex matrices A, the conjugate transpose involution
has the “properness” property
, as required for the existence of the partial order known as the “
-order” on
. More generally, related questions are asked and answered about the ring
of all n × n matrices over an arbitrary associative ring R with any given involution
(as a generalization of the conjugacy map
), yielding a corresponding “
-transposition” involution on
. A criterion is found for this involution of
to be proper, so that
has a corresponding
-order. The special case
is also considered.