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Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. Let $\tilde{\phi}$ be its unique $w^*$-continuous extension on $A^{**}$.

Let us put

$$N_{\phi}:=\{a\in A: \phi(a^*a)=0\}~~~,~~~N_{\tilde{\phi}}:=\{x\in A^{**}: \phi(x^*x)=0\}$$

$N_{\phi}$ forms a closed left ideal in $A$ and $N_{\tilde{\phi}}$ forms a $w^*$-closed left ideal in $A^{**}$. It is obvious that $N_{\phi}\subseteq{N_\tilde{\phi}}$.

Q) Assume that $\phi$ is a pure state. I feel strongly (but have no clear proof) that: $N_{\tilde{\phi}}=\overline{N_{\phi}}^{w^*}$.

Remark. The previous assumption does not hold for any positive linear functional. See the following example:

Example 1: Let $\{x_n\}$ be a dense subset in $[0,1]$ and consider $\phi_j:C([0,1])\to \mathbb{C}$ given by $$\phi_1(f)=\sum \frac{f(x_n)}{2^n}~~~,~~~\phi_2(f)=\int_0^{0.5} f dm $$
where $dm$ is the Lebesgue measure. One may check that $N_{\tilde{\phi_j}}\ne \overline{N_{\phi_j}}^{w^*}$.

It seems that in infinite dimensional commutative $C^*$-algebras the above assertion just holds for pure states however in non-commutative case there are (likely) some non-pure states with this property.

  • $\begingroup$ In commutative algebra it also holds for finite linear combination of characters I think. In $K(H)$ if you take some injective trace class operator it produces a state $\phi$ such that $N_{\overline{\phi}} = 0$ and hence statisfies your property without being pure or finite combination of pure states. $\endgroup$ – Simon Henry Feb 13 '16 at 9:34
  • $\begingroup$ I hav'nt been able to find a satisfying answer, but I have the impression that the situation is very different depending on if in the GNS representation associated to $\phi$ there is or not operator in the image of $A$ (so roughly on whether if algebra is type I/post-liminal or not). My guess would be that it is true for type I/post-liminal algebras and not necessarily for more general algebra... Have you tried to consider non type I examples ? $\endgroup$ – Simon Henry Feb 13 '16 at 12:38
  • $\begingroup$ Everting that I know about: We have always $1-$supp$\phi$=supp$N_{\tilde{\phi}}$ where supp$N_{\tilde{\phi}}$ is the supremum of all positive elements in the unit ball of $N_{\tilde{\phi}}$. Since $\phi$ is a pure state then its support is a minimal projection, say $e$ and so $N_{\tilde{\phi}}=A^{**}(1-e)$ is a maximal $w^*$-closed left ideal in $A^{**}$. It seems that $N_{\phi}=A^{**}(1-e)\cap A$ (which is a maximal closed left ideal in $A$). So the question is : If $e$ is a minimal projection in $A^{**}$ then $$A^{**}(1-e)=\overline{A^{**}(1-e)\cap A}^{w^*}$$ $\endgroup$ – Ali Bagheri Feb 14 '16 at 5:56

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