I would like to continue this discussion.
Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (one can imagine $\mu$ as an arbitrary positive continuous functional on ${\mathcal C}(X)$). We denote by ${\mathcal U}(X)$ the space of all universally integrable functions on $X$.
Nate Eldredge noticed here, that ${\mathcal U}(X)$ is a $C^*$-algebra with respect to the sup-norm: $$ ||f||=\sup_{x\in X}|f(x)|. $$ Question:
Is ${\mathcal U}(X)$ a von Neumann algebra with respect to this norm?