I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the unital case, but I am stuck in the non-unital setting: The authors (N. Brown and N. Ozawa) suggest that one should prove that if $A^{**}$ is semidiscrete then so is $(\tilde{A})^{**}$ and then conclude by proving that if $\tilde{A}$ has the CPAP then so does $A$.

My problem is this: I can't prove that the double dual of the unitization will be semidiscrete. I cannot understand the double dual of the unitization in relevance to the double dual of $A$ at all. The authors state that $(\tilde{A})^{**}\cong A^{**}\oplus\mathbb{C}$ and mention that it is furthermore true that if $B$ is any $C^*$-algebra with a (closed, two-sided) ideal $J$, then $B^{**}\cong J^{**}\oplus(B/J)^{**}$. First of all, does $\cong$ mean as vector spaces or as $C^*$-algebras? How can one prove this isomorphism? Extra bonus question: If all double duals involved are endowed with their ultraweak topologies, is $\cong$ a homeomorphism?


1 Answer 1


Believe it or not, these are $*$-isomorphisms as C${}^*$-algebras. If $J$ is a closed two-sided ideal of $B$ then $J^{**}$ is a weak* closed two-sided ideal of $B^{**}$, and every weak*-closed two-sided ideal of a von Neumann algebra is a direct summand. I suppose these are good exercises. The supremum in $B^{**}$ of an approximate unit for $J$ will be a central projection $p$ such that $pB^{**} = J^{**}$.

Bonus, any $*$-isomorphism of von Neumann algebras is automatically a weak* homeomorphism. That is because it preserves order and hence must be normal.

  • $\begingroup$ Thank you very much for your answer. Much appreciated, I will try the exercises you are pointing out, even though they seem a little difficult to me. $\endgroup$ Aug 25, 2020 at 16:31
  • 1
    $\begingroup$ Hint: a weak*-closed ideal would itself be a von Neumann algebra and hence must have a unit. $\endgroup$
    – Nik Weaver
    Aug 25, 2020 at 21:17

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