On extendability of functionals on Hilbert $C^*$-modules

V. Manuilov

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arXiv · 2205.07089

On extendability of functionals on Hilbert $C^*$-modules

V. Manuilov

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Abstract

Let $M\subset N$ be Hilbert $C^*$-modules over a $C^*$-algebra $A$ with $M^\perp=0$. It was shown recently by J. Kaad and M. Skeide that there exists a non-zero $A$-valued functional on $N$ such that its restriction onto $M$ is zero. Here we show that this may happen even if $A$ is monotone complete. On the other hand, we show that for certain type I $W^*$-algebras this cannot happen.

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V. Manuilov. 2022-05-14. On extendability of functionals on Hilbert $C^*$-modules. https://arxiv.org/abs/2205.07089

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