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A transitivity theorem for algebras of elementary operators

Author: Bojan Magajna
Journal: Proc. Amer. Math. Soc. 118 (1993), 119-127
MSC: Primary 46L05; Secondary 47B48, 47D25
MathSciNet review: 1158004
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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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Keywords: Elementary operators, $ {C^{\ast}}$-algebra, von Neumann algebra
Article copyright: © Copyright 1993 American Mathematical Society

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