Proposition 43.7.4.label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $I \subset M$ be a two-sided ideal, then:
- (1)
$I$ is self-adjoint.
- (2)
Let $x \in I$ with $x > 0$, $E: \cb_{[0, \infty)}\to \text{Proj}(M)$ be its resolution of the identity, and $t > 0$, then $E([t, \infty)) \in I$.
- (3)
If $I$ is weak-operator closed, then there exists a unique projection $z \in Z(M)$ with $I = Mz$.
Proof, [Proposition 2.4.15, AP]. (1): Let $x = u|x|$ be the polar decomposition of $x$, then $|x| = u^{*}x \in I$, and $x^{*} = |x|u^{*} \in I$.
(2): Let $f: [0, \infty) \to \real$ be defined by $f(s) = s^{-1}\one_{\bracsn{s \ge t}}(s)$, then by the Borel functional calculus, $E([t, \infty)) = xf(x) \in I$.
(3): By (5) of Theorem 43.3.1.$\square$
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