I believe I might be the Gleason you are referring to.
I think the paper you might be referring to is one I wrote for a paper about this at an REU I attended two years ago. At the time, I had the goal of making the transition from classical mechanics to quantum mechanics as natural as possible, motivating the axioms of quantum mechanics from those of classical mechanics, which are a lot more intuitive. Unfortunately, given the time constraints of the REU (just a couple of months) and being relatively new to the subject, I found myself mostly following other sources (in particular, Strocchi), which made use of the $C^*$-algebraic formalism. I have to admit, at the time, I did feel a bit uncomfortable throwing away all the unbounded operators, because, having studied "physicist's" quantum mechanics before, these are some things I naturally wanted to include.
Come two years later at the same REU, I tried to tackle the same problem, but this time I wanted to develop the axioms of quantum mechanics so to as allow unbounded operators in the theory. This led me to develop the notion of what I call an $F^*$-algebra ($F$ for Frechet). I am actually still working on the paper at the moment, but, after reading your post, I decided to upload a preliminary copy to my academia.edu account. You should read it, check it out, and see what you think. I'd be very happy to hear any comments you have.
Be warned though, I am still in the process of editing and revising it, so dare I say, there may be some errors. Read it with a skeptical eye and let me know if you catch anything.
Cheers! Jonny Gleason
P.S.: How do you make the accent in Frechet on mathoverflow?