The trace-class of a full Hilbert algebra

Kervin, Michael R. W.

doi:10.1090/S0002-9947-1973-0318900-8

The trace-class of a full Hilbert algebra
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by Michael R. W. Kervin PDF

Trans. Amer. Math. Soc. 178 (1973), 259-270 Request permission

Abstract:

The trace-class of a full Hilbert algebra A is the set $\tau (A) = \{ xy|x \in A,y \in A\}$. This set is shown to be a $\ast$-ideal of A, and possesses a norm $\tau$ defined in terms of a positive hermitian linear functional on $\tau (A)$. The norm $\tau$ is in general both incomplete and not an algebra norm, and is also not comparable with the Hilbert space norm $\left \|\right \|$ on $\tau (A)$. However, a one-sided ideal of $\tau (A)$ is closed with respect to one norm if and only if it is closed with respect to the other. The topological dual of $\tau (A)$ with respect to the norm $\tau$ is isometrically isomorphic to the set of left centralizers on A.

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