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

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 $\mathcal{A}$?

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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.