Definition 44.3.1 (Affiliated).label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $x: D(x) \to H$ be a linear operator, then $x$ is affiliated with $M$, denoted $x \afl M$, if $xy = yx$ for all $y \in M'$[1][2].
The set $\td M$ denotes the family of all closed densely defined operators on $H$ affiliated with $M$.
- In the sense that $D(xy) = D(yx)$ and $xy = yx$ on this domain. keyboard_return
- By Corollary 39.7.5, it is sufficient to check this for unitaries.keyboard_return
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