🚀 go-pugleaf

Say I have a Banach space V.  Now I form the symmetric algebra SV, that
is, all formal (commuting) polynomials in elements of V.  I'd like to
form a Banach algebra completion of SV in some especially nice way.  In
particular, I want the norm on V to coincide with that on S^1V, the 
homogenous polynomials of degree one (a subspace of SV which may be
canonically identified with V).  

This must have been studied, no?  It seems there must be a good way to
do it.  I suppose one could do something like the "universal enveloping
C*-algebra construction," where you start with a *-algebra and give it
a C*-norm which is essentially the sup over all norms of Hilbert space
representations.  This norm has a universal property and in my case too
one could try to define a norm on SV by a universal property -- though
I'd be happier if one could *also* describe the norm a bit more
explicitly.

In any event, what I'd really like is references.

Thanks,
John Baez