Thanks! I will look into these papers and their references. Looks promising!

Yes, but I didn't say much about lower bounds.

Yes, there is a reason :) I'm doing some potential theory on normed spaces, and wanted a couple of concrete examples of spaces which are NOT algebras of functions.

Thanks! The proof of Hölder's inequality one can find in any book, but the other statement: $\tau(xq)\geq 0$ (for all finite projections) iff $x\geq 0$, do you have a reference for it, or is it trivial to prove?

Yes, I do know about the universal property, and that an $S$-inverting homomorphism between to rings give rise to a (unique) homomorphism between the corresponding fraction fields. But, I can't straighten out the details of your last comment; i.e. why $R^{op}S^{-1}$ is isomorphic to $(RS^{-1})^{op}$. I guess that one argues that $(RS^{-1})^{op}$ is a fraction field of $R^{op}$ and then uses universality to equate it to $R^{op}S^{-1}$. But why is $(RS^{-1})^{op}$ is a fraction field of $R^{op}$? Explicit check?

Thanks! How is it that *:R^{op}->RS^{-1} induces *:(RS^{-1})^{op} -> RS^{-1} in your argument?

Thanks! I will have a closer look at your answer later today. You're confirming my suspicion that one needs to use both the left AND right Ore property to prove the fact that (ab)*=ba. As I wanted to learn these things throrougly, I have done all the calculations to check that the set of pairs is actually a division ring (which Ore does not explicitly do in his paper). It was tedious, and sometimes not completely trivial. It would then annoy me if I had to rely on a non-constructive proof to claim that it is a *-algebra :)