On extendability of functionals on Hilbert C∗$C^*$‐modules

Vladimir Manuilov

Mathematische Nachrichten, 2024, vol. 297, issue 3, 998-1005

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

Date: 2024
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https://doi.org/10.1002/mana.202200471

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