Tensor products (and C*-algebras)

[Michael Barr <barr@math.mcgill.ca>]

I think it important to note that tensor products have no universal
mapping property and no categorical definition.  Given an internal hom,
there might or might not be a tensor product that provides a left adjoint
for it.  Given a bifunctor, there might or might not be an internal hom
(or two if the bifunctor is not symmetric) right adjoint to it.

Another point is that the tensor product on abelian groups (or modules
over any commutative right) has two universal mapping properties: It
provides left adjoints for the quite obvious (but still not categorically
defined) internal hom and also represents the functor that takes A to the
functor of bilinear maps out of A x B.

In any case, it makes no sense to ask what is the "right" tensor product.
The right tensor product will be the one that is appropriate to the job
you want to do with it.

Michael