Continuity of positive functionals on topological 1-algebras

It is shown that the set $\text{C}$_{m × n} of complex m × n matrices forms a lower semilattice under the partial ordering A ? B defined by $\text{A}{}^1\text{A = A}{}^1\text{B,}{}^1\text{AA}{}^1\text{= BA}{}^1\text{,}\text{where}\text{A}{}^1$ denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, form = n = 2 and any proper involution 1 of F_{2 × 2}, the corresponding intersections A ∩ B all exist.