Ultraweak continuity of power maps on $W^*$-algebras

Question

Let $\mathcal{A}$ be a $W^*$-algebra. Is the map $a \mapsto a^2$, or more generally the map $a \mapsto a^k$, ultraweakly continuous? (Of course, products are not jointly ultraweakly continuous in general, but products of something with itself are a special case!) If not, does it become so when restricted to the normal/self-adjoint/positive elements of $\mathcal{A}$?

Answer 1

If your function were continuous on selfadjoints, then every bounded ultraweak-convergent net would be ultrastrong convergent. Because if $a_j\to0$ weakly for a net of selfadjoints, the continuity implies that $a_j^2\to0$ weakly and so $a_j\to0$ strongly. For an arbitrary convergent net, since the adjoint is weakly continuous one deduces that the real and imaginary parts converge weakly, and so strongly by the hypothesis.

Of course, the ultraweak and ultrastrong are well-known to be different: for instance, a ultrastrong limit of unitaries is a unitary, while the unitaries are ultraweakly dense in the unit ball.