Exercise 10, § 5

Let E be a complex Hilbert space and A a self-adjoint unital subalgebra of $\mathscr{L}(E)$ which is closed for the topology of simple convergence. (Such an algebra is called a von Neumann algebra.)

a) There exists a subset X of $\mathscr{L}(E)$ such that A coincides with the commutant of X in $\mathscr{L}(E)$. (Use exercise 24 of IV, p. 326.) Conversely, for every self-adjoint subset X, the commutant of X in $\mathscr{L}(E)$ is a self-adjoint unital subalgebra closed for the topology of simple convergence.

b) Let $u\in A$ and let $(j,|u|)$ be the polar decomposition of $u$ (definition 4 of I, p. 140). The elements $j$ and $u$ belong to A.

c) Let $u\in A$ be such that $u$ is normal. For every function $f\in \mathscr{L}_u^{\infty}$(Sp($u$)), the endomorphism $f(u)$ belongs to A.

d) Let $u\in A$. The element $u$ is a divisor of 0 on the right (resp. on the left) in A (cf. A, I, p. 93) if and only if the image of $u$ is not dense in E (resp. if $u$ is not injective). e) Let F denote the set of partial operators on E whose elements are the closed partial operators with dense domain $u$ such that, for every $x$ in X, the partial operator $x\circ u$ is a restriction of the partial operator $u\circ x$. For every $u$ in F and $v$ in A, the partial operators $u^*,u+v,v\circ u$ and $u\circ v$ belong to F.

f) Let $u\in F$ and let $(j,|u|)$ be the polar decomposition of $u$ (definition 7 of IV, p. 290). The partial operator $|u|$ belongs to F and $j$ belongs to A.

g) Let $u\in F$ be such that $u$ is normal. For every function $f\in \mathscr{L}_u$(Sp($u$)), the partial operator $f(u)$ belongs to F; if $f$ is bounded, then $f(u)$ belongs to A.

h) Let $u\in A$ be injective and have dense image in E. The partial operator $u^{-1}$ belongs to F.

i) For every $u$ in F, there exist elements $u_1$ and $u_2$ of A such that $u_2$ is injective and has dense image and $u=u_1u^{-1}_2$. (Reduce first to the case where $u$ is self-adjoint and positive, then apply the universally measurable functional calculus to the functions $f_1:t\mapsto t\varphi (t) +\psi (t)$ and $f_2:t\mapsto \varphi (t) +t^{-1}\psi (t)$, where $\varphi$ and $\psi$ are the characteristic functions of the intervals $[0,1[$ and $[1,+\infty [$ respectively.)

j) Suppose that every injective element of A with dense image is surjective. Then the algebra A satisfies Ore's condition: whatever $u$ and $s$ in A may be, if $s$ is not a divisor of 0 (neither on the right nor on the left), then there exist $u'$ and $s'$ in A such that $s'$ is not a divisor of 0 and $us'=su'$.

k) The conclusion of the preceding question is not always valid. (Consider $A =\mathscr{L}(E).$)