On linear functionals on Clifford modules and their extensions
type: Item
instance of: scholarly article
title: On linear functionals on Clifford modules and their extensions
zbMATH Open document ID: 1361.15031
DOI: 10.1134/S0081543816040076
author: I. A. Borovikov
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2016-11-07T00:00:00Z
review text: 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.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 46A22
zbMATH DE Number: 6648143
zbMATH Keywords: Clifford modules
zbMATH Keywords: linear functionals
zbMATH Keywords: Clifford algebra
zbMATH Keywords: Hahn-Banach theorem
MaRDI profile type: Publication
cites work: Extensions of functionals on complex linear spaces
cites work: Q3729194
cites work: Q3241211
cites work: Q5588717
cites work: Matrix representations of Clifford algebras
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description: scientific article; zbMATH DE number 6648143