On linear functionals on Clifford modules and their extensions
Let \((~,~)\) be the standard inner product on the vector space \(\mathbb{R}^{n} \), and \(A_{n}\) be the \(2^{n}\)-dimensional Clifford algebra on \(\mathbb{R}^{n}\) corresponding to the bilinear symmetric form \(-(~,~)\). For example, \(A_{0}=\mathbb{R}\), \(A_{1}=\mathbb{C}\) and \(A_{2}=\mathbb{H}\)\ (the quaternions). The author studies the category \(A_{n}\)-mod of left unital \(A_{n}\)-modules. In particular, let \((~)_{\mathbb{R}}:A_{n}\)-mod \(\rightarrow\) \(\mathbb{R}\)-mod be the forgetful functor and let \(M^{\ast}\) denote the dual of an \(A_{n}\)-module \(M\). The author generalizes the well-known case \(n=1\) (where \(A_{n}=\mathbb{C}\)) to show that the functors \((~)_{\mathbb{R}} \circ(~)^{\ast}\) and \((~)^{\ast}\circ(~)_{\mathbb{R}}\) are isomorphic. Furthermore, if \(M\) is an \(A_{n}\)-module and \(M_{1}\subseteq M\) is a submodule, then each functional \(f\in M_{1}^{\ast}\) can be extended to a functional in \(M^{\ast}\). It is then claimed that the following analogue of the Hahn-Banach theorem is true. (Proposition 3): If \(p\) is a seminorm on \(M_{\mathbb{R}}\) such that \(p(ax)\leq\left| a\right| _{A_{n}}p(x)\) for all \(x\in M\) and \(a\in A_{n}\) and \(M_{1}\) is an \(A_{n}\)-submodule of \(M\) then, for each \(f\in M_{1}^{\ast}\) satisfying \(\left| f(x)\right| _{A_{n}}\leq p(x)\) for all \(x\in M_{1}\) there exists an extension \(\tilde{f}\in M^{\ast}\) such that \(\left| \tilde{f}(x)\right| _{A_{n}}\leq\sqrt{K_{n}}p(x)\). However, as the author notes it is not clear what the appropriate value of \(K_{n}\) is (in general \(K_{n}\) cannot be \(1\)). This makes it difficult to interpret this result, and the proof of Propostion 3 in the translation is problematic.
scientific article; zbMATH DE number 6648143
scientific article