# Is the bitranspose continuous for the $\sigma$-strong topology?

Let $$\varphi\colon A\to B$$ be a bounded, linear map between C*-algebras. Is the bitranspose $$\varphi^{**}\colon A^{**}\to B^{**}$$ continuous when the von Neumann algebras $$A^{**}$$ and $$B^{**}$$ are equipped with their $$\sigma$$-strong topologies?

Motivation/Background: Note that $$\varphi^{**}$$ is clearly continuous when $$A^{**}$$ and $$B^{**}$$ are equipped with their $$\sigma$$-weak topologies, since these agree with the weak$${}^*$$-topology from the preduals, and $$\varphi^{**}$$ is weak$${}^*$$-continuous (that is, $$\sigma(A^{**},A^*)-\sigma(B^{**},B^*)$$-continuous).

If $$\varphi$$ if completely positive, then it follows that $$\varphi^{**}$$ is a completely positive, normal map, and therefore is continuous for the $$\sigma$$-strong topologies. Thus, the question is only interesting if $$\varphi$$ is not completely positive.

On bounded sets of a von Neumann algebra, the $$\sigma$$-strong topology agrees with the strong (operator) topology (SOT). If $$M\subseteq B(H)$$ is a von Neumann algebra, then a net $$(a_j)_j$$ in $$M$$ SOT-converges to $$a\in M$$ if $$\|a_j\xi-a\xi\|\to 0$$ for every $$\xi\in H$$.

• Do you know it's true if $A$ and $B$ are abelian? Mar 28 at 14:09
• @NikWeaver Good point. I don't even know the answer in that case. Mar 28 at 14:19
• Eh ... I'm not sure $\sigma$-weak and $\sigma$-strong are even different in the abelian case ... Mar 28 at 14:26

Let $$K(H)$$ be the compacts on a separable infinite dimensional Hilbert space with orthonormal basis $$\{ e_n \}.$$ Let $$T:K(H)\rightarrow K(H)$$ be the transpose map (i.e. $$T(e_{n,m})=e_{m,n}$$ on matrix units). Then $$T$$ is $$\sigma$$-weakly continuous so $$T^{**}:B(H)\rightarrow B(H)$$ will also be the transpose map. Let $$S$$ be the shift $$Se_n=e_{n+1}.$$ Then $$T(S^{*n})=S^n$$ for all $$n.$$ Finally $$S^{*n}\rightarrow 0$$ strongly while $$T(S^{*n})=S^n$$ does not converge strongly to anything.
Since all of these maps $$S^n, S^{*n}$$ are in the unit ball, as you mention the strong and $$\sigma$$-strong topologies coincide.