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Representations of a class of real $B^ *$-algebras as algebras of quaternion-valued functions


Author: S. H. Kulkarni
Journal: Proc. Amer. Math. Soc. 116 (1992), 61-66
MSC: Primary 46K05; Secondary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1992-1110546-4
MathSciNet review: 1110546
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Abstract: For a compact Hausdorff space $X$, let $C(X,{\mathbf {H}})$ denote the set of all quaternion-valued functions on $X$. It is proved that if a real ${B^*}$-algebra $A$ satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in $A$ is normal, then $A$ is isometrically $*$-isomorphic to a closed $*$-subalgebra of $C(X,{\mathbf {H}})$ for some compact Hausdorff $X$. In particular, a real ${C^*}$-algebra in which every element is normal is isometrically $*$-isomorphic to a closed $*$-subalgebra of $C(X,{\mathbf {H}})$.


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Article copyright: © Copyright 1992 American Mathematical Society