Normed Rings and Spectral Representation

Abstract

A linear space A over a scalar field (F) is said to be an algebra or a ring over (F), if to each pair of elements x, y ∈ A a unique product xy ∈ A is defined with the properties:

$$\left. {\begin{array}{*{20}{c}} {(xy)z = x(yz)\;(associativity),} \\ {x(y + z) = xy + xz\;(distributivity),} \\ {\alpha \beta (xy) = (\alpha x)(\beta y)} \\ \end{array} } \right\}$$

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