Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of trace-class operators, one can show using standard result of Banach space theory that, any bounded linear functional $\phi$ on $B(\mathcal{H})$ can be approximated in weak$^\*$ topology by (bounded) trace-class operators. In other words, $\phi$ is approximated by normal linear functionals on $B(\mathcal{H})$. My question is the following:


If the linear functional $\phi$ is **positive**, can $\phi$ be approximated by **positive** normal linear functionals in weak$^\*$ topology?


Moreover, can this be generalized to completely positive maps? The topology here in consideration is bounded-weak topology. More specifically, if $M$ is a von Neumann algebra, then can every completely positive map $\Phi:M\to B(\mathcal{H})$ be approximated by normal completely positive maps in bounded-weak topology?


Some reference would be appreciated on these topics as I am new to them. Thank you.



Are (completely) positive maps approximated by normal (completely) positive maps?
Also the answer to the second question is **yes**, and the approximation may be chosen to converge in the point-ultrastrong$^\*$ topology.


First, by choosing a net of finite rank orthogonal projections $p\_i \in B(\mathcal{H})$ such that $p\_i \rightarrow 1$ strongly, the completely positive maps $\Phi\_i : M \rightarrow B(p\_i H) : \Phi\_i(a) = p\_i \Phi(a) p\_i$ converge to $\Phi$ in the point-ultrastrong$^\*$ topology. So it suffices to deal with completely positive maps $\Phi : M \rightarrow M\_n(\mathbb{C})$. This can be found in [BO, Corollary 1.6.3]. By [BO, Proposition 1.5.14],
$$\omega : M\_n(\mathbb{C}) \otimes M \rightarrow \mathbb{C} : \omega(A) = \sum\_{i,j} \Phi(A\_{ij})\_{ij}$$
is a positive functional. Choose a net $\omega\_k$ of normal positive functionals on $M\_n(\mathbb{C}) \otimes M$ that converge pointwise to $\omega$. Again by [BO, Proposition 1.5.14], there is a corresponding net of completely positive maps
$$\Phi\_k : M \rightarrow M\_n(\mathbb{C}) : (\Phi\_k(a))\_{ij} = \omega\_k(e\_{ij} \otimes a) \; .$$
By construction, the maps $\Phi\_k$ are normal and they converge to $\Phi$ in the point-norm topology.


[BO] N.P. Brown and N. Ozawa, C$^\*$-algebras and finite-dimensional approximations.
*Graduate Studies in Mathematics* **88**. American Mathematical Society, Providence, 2008.


