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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.