Dear Peter,
thanks for writing back! I think you mean a linear isomorphism V->Hom(\bar V,C). Although Hom(\bar V,C) and Hom(V,\bar C) are the same set, they are not the same Hilbert space, because the scalar multiplication is skewed on one relative to the other. So an inner product either amounts to a skew linear map V->Hom(V,\bar C), or to a linear map V->Hom(\bar V,C).
My understanding of bar is the following: a map of C-vector spaces V->W is skew-linear iff either \bar V-> W or (what amounts to the same) V -> \bar W is linear. Therefore, the two C-vector spaces Hom(\bar V,C) and Hom(V,\bar C) have indeed the same underlying set of elements (namely skew-linear functionals on V). The additive structure is the same too, but the multiplicative structure is different, since the scalars act on functionals in the target; therefore, we get Hom(\bar V, C)= \bar(Hom(V, \bar C)). An inner product on V is classically either a skew-linear map V -> Hom(V,C) or a linear map V -> Hom(\bar V,C) with Hermitian symmetry. What I wanted initially is an isomorphism V-> Hom(\bar V,C), so my first description of the involutive structure on Banachspaces (V|-> Hom(V,\bar C)) was wrong and should be replaced by V|->Hom(\bar V,C).
So unless I am mistaken, the definition of the functor "bar", and of the "complex conjugate" of a map f:V->W, cannot be reduced to the existence of just one object \bar C. you are completely right !
I certainly agree with you that there exists an involutive (in your sense) structure on the category FVect (I'll again restrict myself to the finite dimensional case) so that its dagger core is (dagger equivalent to) FHilb. (Well, almost: this is true if we drop the condition of positivity from the definition of inner product).
this also is right. I wonder whether there are some purely categorical properties which can distinguish positive definiteness at least from mixed signature.
But my point was that it can't be defined unless you first fix arbitrary isomorphisms V -> \bar V.
Am I right?
after the above yes.
What is the center of a pivotal category? You mean, where the objects are pairs (A,e) of an object and a natural transformation e_B : A * B -> B * A, subject to coherence w.r.t. B=B'*B'' and B=I?
yes, precisely. This is the categorical analog of the Drinfeld double of a Hopf algebra. All the best, Clemens.