The structure of quasimultipliers of 𝐶*-algebras

Lin, Hua Xin

doi:10.1090/S0002-9947-1989-0937248-3

The structure of quasimultipliers of $C^ *$-algebras
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by Hua Xin Lin

Trans. Amer. Math. Soc. 315 (1989), 147-172

DOI: https://doi.org/10.1090/S0002-9947-1989-0937248-3

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Abstract:

Let $A$ be a ${C^\ast }$-algebra and ${A^{\ast \ast }}$ its enveloping ${W^\ast }$-algebra. Let ${\text {LM}}(A)$ be the left multipliers of $A$, ${\text {RM}}(A)$ the right multipliers of $A$ and ${\text {QM}}(A)$ the quasi-multipliers of $A$. A question was raised by Akemann and Pedersen [1] whether ${\text {QM}}(A) = {\text {LM}}(A) + {\text {RM}}(A)$. McKennon [20] gave a nonseparable counterexample. L. Brown [6] shows the answer is negative for stable (separable) ${C^\ast }$-algebras also. In this paper, we mainly consider $\sigma$-unitial ${C^\ast }$-algebras. We give a criterion for ${\text {QM}}(A) = {\text {LM}}(A) + {\text {RM}}(A)$. In the case that $A$ is stable, we give a necessary and sufficient condition for ${\text {QM}}(A) = {\text {LM}}(A) + {\text {RM}}(A)$. We also give answers for other ${C^\ast }$-algebras.

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