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Representation of a real $ B\sp *$-algebra on a quaternionic Hilbert space

Author: S. H. Kulkarni
Journal: Proc. Amer. Math. Soc. 121 (1994), 505-509
MSC: Primary 46K05; Secondary 46H15, 46K10
MathSciNet review: 1186133
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Abstract: Let A be a real $ {B^ \ast }$-algebra containing a $ \ast $-subalgebra that is $ \ast$-isomorphic to the real quaternion algebra $ \mathbb{H}$. Suppose the spectrum of every self-adjoint element in A is contained in the real line. Then it is proved that there exists a quaternionic Hilbert space X and an isometric $ \ast $-isomorphism $ \pi $ of A onto a closed $ \ast $-subalgebra of $ BL(X)$, the algebra of all bounded linear operators on X. If, in addition to the above hypotheses, every element in A is normal, then A is also proved to be isometrically $ \ast $-isomorphic to $ C(Y,\mathbb{H})$, the algebra of all continuous $ \mathbb{H}$-valued functions on a compact Hausdorff space Y.

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Article copyright: © Copyright 1994 American Mathematical Society

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