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Subspaces of \(C^ *\)-algebras. (English) Zbl 0646.46055
Let V be a vector space over the complex numbers C and let \(M_ n(V)=V\otimes M_ n(C)\) denote the vector space of \(n\times n\) matrices \([v_{ij}]\) with entries in V. Let \(A[v_{ij}]=[\sum_{k}a_{ik}v_{kj}]\) and \([v_{ij}]A-[\sum a_{kj}v_{ik}]\) for \([v_{ij}]\) in \(M_ n(V)\) and \(A=[a_{ij}]\) in M(C). If \(v\in M_ n(V)\) and \(\omega \in M_ m(V)\) let \(v\oplus \omega\) be identified with the \(2\times 2\) matrix in \(M_{n+m}(V)\) with 0 off the diagonal and v and \(\omega\) down the diagonal. Then \((V,\{\| \|_ n\})\) is said to be a matricially normed space if each \(\| \|_ n\) is a norm on the vector space \(M_ n(V)\) that satisfies the properties (I) \(\| v\oplus 0\|_{n+m}=\| v\|_ n\) for \(v\in M_ n(V)\) and \(0\in M_ n(V)\) and (II) \(\| Bv\|_ n\leq \| B\| \| v\|_ n\) and \(\| vB\|_ n\leq \| B\| \| v\|_ n\) for \(B\in M_ n(V)\), \(v\in M_ n(V)\), and for every \(n,m=1,2,... \). A linear map T of the vector space V into the vector space W induces a linear map \(T_ ntheory\) viewpoint. As a complement of the existing textbooks in electrodynamics, this treatment not only gives us background information about the idea of gauge theory but also deepens our understanding of electromagnetism.

MSC:
46L05 General theory of \(C^*\)-algebras
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