×

Double commutants of multiplication operators on \(\mathcal{C}(K)\). (English) Zbl 1342.47048

Let \(K\) be a compact Hausdorff space and denote by \( \mathcal C (K)\) the space of all real or complex valued continued functions on \(K\). For \(f \in \mathcal C (K)\), denote by \(M_f\) the multiplication operator on \(\mathcal C(K)\) implemented by \(f\).
The author investigates an analogue of von Neumann Double Commutant Theorem on \(\mathcal C (K)\). More precisely, in the present paper, \(K\) is said to have the double commutant property (\(K \in \mathcal {DCP}\)) if for any \(f \in \mathcal C(K)\), the double commutant of \(M_f\) is the closure in the weak operator topology of the algebra generated by \(M_f\) and the identity operator \(I\). In [Positivity 14, No. 4, 753–769 (2010; Zbl 1218.47053)], the author proved that if \(K\) is a locally connected continuum, then \(K \in \mathcal {DCP}\). In the paper under review, he gives examples of compact spaces which are not locally connected but are in \(\mathcal {DCP}\).

MSC:

47B38 Linear operators on function spaces (general)
47B48 Linear operators on Banach algebras

Citations:

Zbl 1218.47053
PDFBibTeX XMLCite
Full Text: arXiv