A transitivity theorem for algebras of elementary operators

Magajna, Bojan

doi:10.1090/S0002-9939-1993-1158004-6

A transitivity theorem for algebras of elementary operators
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by Bojan Magajna

Proc. Amer. Math. Soc. 118 (1993), 119-127

DOI: https://doi.org/10.1090/S0002-9939-1993-1158004-6

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Abstract:

Let $\mathcal {A}$ be a ${C^{\ast }}$-algebra and $\mathcal {E}$ the algebra of all elementary operators on $\mathcal {A}$, and let $\vec a = ({a_1}, \ldots ,{a_n}),\;\vec b = ({b_1}, \ldots ,{b_n}) \in {\mathcal {A}^n}$. It is proved that $\vec b$ is contained in the closure of the set $\{ (E{a_1}, \ldots ,E{a_n}):E \in \mathcal {E}\}$ if and only if each complex linear combination $\sum \nolimits _{j = 1}^n {{\lambda _j}} {b_j}$ is contained in the closed two-sided ideal generated by $\sum \nolimits _{j = 1}^n {{\lambda _j}} {a_j}$. In particular, a bounded linear operator on $\mathcal {A}$ preserves all closed two-sided ideals if and only if it is in the strong closure of $\mathcal {E}$.

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