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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.

46L05 General theory of \(C^*\)-algebras
Full Text: DOI
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