Geodesic
Dense Subalgebras of Left Hilbert Algebras
Canadian journal of mathematics, Tome 34 (1982) no. 6, pp. 1245-1250

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Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]). 
Daele, A. van. Dense Subalgebras of Left Hilbert Algebras. Canadian journal of mathematics, Tome 34 (1982) no. 6, pp. 1245-1250. doi: 10.4153/CJM-1982-086-x
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