A characterization of minimal left or right ideals of matrix algebras

The algebra of m×m matrices over real numbers, complex numbers or quaternions is equipped with the standard inner product Re tr(xy ^*) Work done in part during the Summer Research Institute of the Canadian Mathematical Congress, 1971.). It is easy to see that the left ideals of this algebra have the property that the projections of unitary matrices on them have constant length. of course, the right ideals have the same property. This property and a condition on dimension give a characterization of minimal left or right ideals of this algebra. We use this characterization to determine all orthogonal transformations of this algebra which preserve unitary matrices.