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