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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 $p(e_\lambda)<K$ 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.

Thanks for your time.

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