A criterion for the properness of star-transposition as an involution on n × n matrices over any associative star-ring

Abstract

On the ring M n ( C ) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the “properness” property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the “ * -order” on M n ( C ) . More generally, related questions are asked and answered about the ring M n ( R ) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding “ * -transposition” involution on M n ( R ) . A criterion is found for this involution of M n ( R ) to be proper, so that M n ( R ) has a corresponding * -order. The special case R = Z k is also considered.