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## Studia Mathematica

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## The order topology on duals of $C^\ast$-algebras and von Neumann algebras

### Tom 254 / 2020

Studia Mathematica 254 (2020), 219-236 MSC: Primary 46L10, 46L05; Secondary 46L30, 06F30. DOI: 10.4064/sm190108-11-7 Opublikowany online: 21 February 2020

#### Streszczenie

For a von Neumann algebra $\mathcal M$, we study the order topology associated to the hermitian part $\mathcal M_*^s$, and to intervals of the predual $\mathcal M_*$. It is shown that the order topology on $\mathcal M_*^s$ coincides with the topology induced by the norm. In contrast, it is proved that the condition of having the order topology, associated to the interval $[0,\varphi ]$, equal to the topology induced by the norm, for every $\varphi \in \mathcal M_*^+$, is necessary and sufficient for the commutativity of $\mathcal M$. It is also proved that if $\varphi$ is a positive bounded functional on a $C^\ast$-algebra $\mathcal A{}$, then the norm-null sequences in $[0,\varphi ]$ coincide with the null sequences, with respect to the order topology on $[0,\varphi ]$, if and only if the von Neumann algebra $\pi _\varphi (\mathcal A)’$ is of finite type (where $\pi _\varphi$ denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology, on order-bounded parts of dual spaces, are inequivalent for all $C^\ast$-algebras that are not of type I.

#### Autorzy

• Emmanuel ChetcutiDepartment of Mathematics
Faculty of Science
University of Malta
Msida MSD 2080, Malta
e-mail
• Jan HamhalterDepartment of Mathematics
Faculty of Electrical Engineering
Czech Technical University in Prague
Technická 2, 166 27 Praha 6, Czech Republic
e-mail

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