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Proceedings of the American Mathematical Society

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Trace-class and centralizers of an $ H\sp{\ast} $-algebra

Author: Parfeny P. Saworotnow
Journal: Proc. Amer. Math. Soc. 26 (1970), 101-104
MSC: Primary 46.60
MathSciNet review: 0267403
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Abstract: Let $ A$ be a proper $ {H^ \ast }$-algebra. Let $ \tau (A) = \{ xy\vert x,y \in A\} $, let $ R(A)$ be the set of all bounded linear operators $ S$ on $ A$ such that $ S(xy) = (Sx)y$ for all $ x,y \in A$ and let $ C(A)$ be the closed subspace of $ R(A)$ generated by the operators of the form $ La:x \to ax,a \in A$. It is shown that $ \tau (A)$ can be identified with the space of all bounded linear functionals on $ C(A)$ and that $ R(A)$ is the dual of $ \tau (A)$. Also it is proved that $ \tau (A)$ is a Banach algebra.

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Keywords: Trace-class, $ {H^ \ast }$-algebra, dual, centralizer, right centralizer, bounded linear functional
Article copyright: © Copyright 1970 American Mathematical Society