# Can I construct a monoidal category on locally convex modules over algebras with approximate identiy

I am faced with the following problem: Let $$A$$ be a complete locally convex algebra with a uniform approximation of identity, that is a net $$e_\lambda$$, such that $$p(e_\lambda a-a)\rightarrow 0$$ for all $$p$$ and there is a constant $$K>0$$ with $$p(e_\lambda) for all $$p$$. Furthermore, consider a complete locally convex left $$A$$-module $${}_AM$$, that is essential, i.e. $$\operatorname{span} \left( AM\right) ^{ \operatorname{cl} } =M$$. (everything is in general non-Féchet) I want to show (or refute), that the canonical mapping $$\psi :\sum_i a_i \otimes m_i \mapsto \sum_i a_i ,m_i$$ gives a linear homeomorphism $$A\otimes_A M\simeq M$$ (I consider the $$A$$-balanced tensor product equipped with the projective cross norm and completed). I was able to proof continuity and injectivity very easily, however I don't see a way to proof neither surjectivity nor openness.

From the literature I now that in the Fréchet case, $$e_{ \lambda }$$ even becomes an approximation of unity in $$M$$ (Robert S. Doran, Josef Wichmann - Approximate Identities and Factorization in Banach Modules Proposition 25.3). I am pretty sure this still holds for the non-Fréchet case.

My idea was to take a sequence converging in the image of $$\psi$$ and then show, that the sequence of preimages is a cauchy sequence for the projective norm, so far unfruitful though.