re: quantum logic

[Michael Barr <barr@barrs.org>]

I think Rick Blute (+ collaborators) has done some things with this. It is
not clear whether you want a self-duality or a *-autonomous category.  If
you stick to finite dimensional Hilbert spaces, the situation seems
simple.  If V and W are inner product spaces, then for f, g: V --> W, let
f.g = \sum f(v_i).g(v_i) the sum taken over an orthonormal basis.  I
believe this is invariant to an orthonormal base change and it is
obviously positive definite.  For infinite dimensional spaces, you would
have to stick to f for which \sum f(v_i)^2 < oo.  But this isn't a
category.  It is closed under composition (I think) but certainly lacks
identities.  This gives rise to something called a nuclear category.  The
category has all maps and there is sub-non-category of nuclear maps.  This
all goes back (needless to say) to Grothendieck.

If by *-category you just mean self dual, well then Hilbert spaces
certainly are that.  Self dual categories are a dime a dozen.  Just take C
x C^op.  The amazing thing is that if C is closed, C x C^op is
*-autonomous, (assuming C has binary cartesian products).

Michael