I have no intuition in this field so excuse me if this is trivial.

Let $X$ be a compact Hausdorff space, and $C(X)$ the algebra of continuous functions on $X$ with the usual $\sup$-norm. This is a $C^*$-algebra, and each state $\omega:C(X)\to{\mathbb C}$ generates a GNS-representation
$$
\pi_\omega: C(X)\to B(H_\omega).
$$
Let us consider the *$\sigma$-strong** topology on $B(H_\omega)$, i.e. the topology defined in the Bratteli–Robinson book Operator Algebras and Quantum Statistical Mechanics 1 by the seminorms of the form
$$
A\mapsto \sqrt{\sum_{n=1}^\infty \lVert A\xi_n\rVert^2+\sum_{n=1}^\infty \lVert A^*\xi_n\rVert^2}
$$
where $\xi_n$ are various sequences in $H_\omega$ with the property
$$
\sum_{n=1}^\infty \lVert\xi_n\rVert^2<\infty.
$$
Let $\tau$ denote this $\sigma$-strong* topology on $B(H_\omega)$, and $B_\tau(H_\omega)$ denote this space with this topology. So we have a (continuous) mapping
$$
\pi_\omega: C(X)\to B_\tau(H_\omega).
$$
Each state $\omega:C(X)\to{\mathbb C}$ generates such a mapping, and together these mappings generate a topology on the linear space $C(X)$, i.e. the initial topology on $C(X)$ generated by the family of mappings $\pi_\omega$. And since $\tau$ are locally convex topologies on $B(H_\omega)$, and $\pi_\omega$ are linear mappings, the corresponding initial topology on $C(X)$ is locally convex. My question is the following:

What is the completion of $C(X)$ with respect to the initial topology generated by the mappings $\pi_\omega$?

I have a suspicion that this must be the space $U(X)$ of universally integrable functions (see Leader - On Universally Integrable Functions), but up to now I did not manage to prove it. Is this true?

Actually, if we replace this $\sigma$-strong* topology with the other standard operator topologies on the spaces $B(H_\omega)$, the corresponding completions of $C(K)$ are also interesting for me. I would be grateful if somebody could cast a light on this.

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