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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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##### References:
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