A separably acting abelian von Neumann algebra has a self-adjoint generator

By [F1], choose a dense sequence $(x_n{)}_{n\in \mathbb{N}}$ in $H$. On the operator unit ball ${\mathcal{B}}_1(H)$ consider the coefficient map $\mathrm{\Phi }(T)=(⟨T{x}_n,x_m⟩{)}_{(n,m)\in {\mathbb{N}}^2}\mathrm{.}$ Its coordinates are WOT-continuous by [F2]. Conversely, they induce the WOT on this ball: if $T,U\in {\mathcal{B}}_1(H)$, then $\mathrm{\parallel }U-T\mathrm{\parallel }\le 2$, and for dense sequence approximants $x_i\to x$, $x_j\to y$, $\mathrm{\mid }⟨(U-T)x,y⟩-⟨(U-T)x_i,x_j⟩\mathrm{\mid }\le 2\mathrm{\parallel }x-x_i\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }+2\mathrm{\parallel }{x}_i\mathrm{\parallel }\mathrm{\parallel }y-x_j\mathrm{\parallel }\mathrm{.}$ Thus every finite matrix-coefficient neighborhood can be refined by one involving only the countable family of coordinates of $\mathrm{\Phi }$.

Fix $k\ge 1$. For a digit string $\eta =({\eta }_1,\dots ,{\eta }_k)\in \{0,1{\}}^k$, set $s_{\eta }:={\sum }_{j=1}^k{3}^{-j}{\eta }_j,I_{\eta }:=[s_{\eta },s_{\eta }+\frac{1}{2}{3}^{-k}]\mathrm{.}$ These finitely many intervals lie in $[0,1\mathrm{/}2]$ and are pairwise separated. Indeed, if two strings first differ at index $\mathrm{\ell }\le k$, the difference of their left endpoints has absolute value at least $3^{-\mathrm{\ell }}-{\sum }_{j=\mathrm{\ell }+1}^k{3}^{-j}=\frac{1}{2}{3}^{-\mathrm{\ell }}+\frac{1}{2}{3}^{-k};$ subtracting their common interval length $\frac{1}{2}{3}^{-k}$ leaves a gap at least $\frac{1}{2}{3}^{-\mathrm{\ell }}>0$. Assign value ${\eta }_k$ on $I_{\eta }$; define $f_k$ linearly across each gap between consecutive intervals, with endpoint values inherited from those intervals, and constantly on each exterior interval with the value at the adjacent endpoint. This defines a continuous real piecewise-linear function on $[-1\mathrm{/}2,1\mathrm{/}2]$. Its values on $[0,1\mathrm{/}2]$ are thus $0$ or $1$ on the cylinder intervals according to the $k$-th digit.

By [F3, F4], the product ${\mathbb{C}}^{{\mathbb{N}}^2}$ is second countable. Pulling its countable basis back by $\mathrm{\Phi }$ gives a countable basis for ${\mathcal{B}}_1(H)$, because step 1.1 shows that its initial topology is the relative WOT. Its subspace ${\mathcal{A}}_1:=\{T\in \mathcal{A}:\mathrm{\parallel }T\mathrm{\parallel }\le 1\}$ is second countable by [F4]. Fix a countable basis of ${\mathcal{A}}_1$.

For each nonempty member of this basis, AC [F11] selects one operator in that member; for an empty member put $0$. Enumerating the countable basis, and repeating an entry if it is finite, gives a sequence $(T_j{)}_{j\in \mathbb{N}}$ WOT-dense in ${\mathcal{A}}_1$. Indeed every nonempty relatively open subset contains a basis member and hence a selected point.

For $j\in \mathbb{N}$ define $X_{j,0}:=\frac{T_j+T_j^{\ast }}{2},X_{j,1}:=\frac{T_j-T_j^{\ast }}{2i}\mathrm{.}$ Because $\mathcal{A}$ is a star-subalgebra, both lie in $\mathcal{A}$; [F6, F12] show they are self-adjoint contractions, and $T_j=X_{j,0}+i{X}_{j,1}$. All these operators commute because $\mathcal{A}$ is abelian.

For each $X=X_{j,\tau }$ and real threshold $q$, let $P_X(q):=E_X((q,\mathrm{\infty }))\mathrm{.}$ This projection belongs to $\mathcal{A}$. To see this directly, on $\sigma (X)\subseteq [-1,1]$ set $g_m(t):=\min \{1,m\max \{t-q,0\}\}\mathrm{.}$ Each $g_m$ is continuous, lies between $0$ and $1$, and converges pointwise to ${\mathbf{1}}_{(q,\mathrm{\infty })}$, including value $0$ at $t=q$. By [F8], $g_m(X)$ lies in the norm closure of the unital star-algebra generated by $X$, hence in $\mathcal{A}$: norm convergence implies WOT convergence and $\mathcal{A}$ is WOT-closed by [F5], [F9]. By [F9], $g_m(X)$ converges strongly, therefore weakly, to $P_X(q)$; WOT-closedness gives $P_X(q)\in \mathcal{A}$. The spectral bound $\sigma (X)\subseteq [-1,1]$ is [F7].

For $n\in \mathbb{N}$ and $0\le \mathrm{\ell }\le 2^n$ put $r_{n,\mathrm{\ell }}:=-2+4\mathrm{\ell }{2}^{-n}\mathrm{.}$ The projections $P_{X_{j,\tau }}(r_{n,\mathrm{\ell }})$, indexed by $(j,\tau ,n,\mathrm{\ell })$, form an explicitly countable family: repeated pairing from [F4] enumerates ${\mathbb{N}}^4$, and invalid $\tau$ or $\mathrm{\ell }$ indices may be assigned the zero projection. List the family, allowing repetitions, as $(P_k{)}_{k\ge 1}$. Every $P_k$ is in $\mathcal{A}$ by step 5.1, so these projections commute pairwise.

Fix $X=X_{j,\tau }$. For each $n$, the intervals $(r_{n,\mathrm{\ell }-1},r_{n,\mathrm{\ell }}]$, $1\le \mathrm{\ell }\le 2^n$, partition $(-2,2]$ and cover $\sigma (X)$. Their spectral projections are $Q_{n,\mathrm{\ell }}=P_X(r_{n,\mathrm{\ell }-1})-P_X(r_{n,\mathrm{\ell }})\mathrm{.}$ The step function taking value $r_{n,\mathrm{\ell }-1}$ on this interval differs from the identity function by at most $4\cdot 2^{-n}$ on $\sigma (X)$. The norm formula for the Borel calculus in [F9] therefore gives $\parallel X-{\sum }_{\mathrm{\ell }=1}^{2^n}{r}_{n,\mathrm{\ell }-1}{Q}_{n,\mathrm{\ell }}\parallel \le 4\cdot 2^{-n}\mathrm{.}$ Each approximant is a finite linear combination of the listed projections, so every $X_{j,\tau }$ is a norm, hence WOT, limit of such combinations.

Define $S_N:={\sum }_{k=1}^N{3}^{-k}{P}_k,S:={\sum }_{k=1}^{\mathrm{\infty }}{3}^{-k}{P}_k\mathrm{.}$ By [F9], $\mathrm{\parallel }{P}_k\mathrm{\parallel }\le 1$. Since ${\sum }_{k\ge 1}{3}^{-k}=1\mathrm{/}2$, the series converges in operator norm. Each $S_N$ is self-adjoint and belongs to $\mathcal{A}$; its norm limit $S$ is self-adjoint and belongs to $\mathcal{A}$ by WOT-closedness. Also $\mathrm{\parallel }{S}_N\mathrm{\parallel },\mathrm{\parallel }S\mathrm{\parallel }\le 1\mathrm{/}2$.

Let $\mathcal{C}$ be the unital star-algebra generated by $(P_k)$. It is contained in $\mathcal{A}$, and its WOT closure is contained in $\mathcal{A}$ by [F5]. The WOT closure of a linear subspace is linear because addition and scalar multiplication are WOT-continuous. Step 7.1 puts every $X_{j,\tau }$ in that closure, hence every $T_j=X_{j,0}+i{X}_{j,1}$ there. Since $(T_j)$ is WOT-dense in ${\mathcal{A}}_1$ by step 3.1, the closure contains ${\mathcal{A}}_1$. Every $T\in \mathcal{A}$ is a scalar multiple of an element of ${\mathcal{A}}_1$ (with $T=0$ immediate), so ${\bar{\mathcal{C}}}^{\mathrm{W}\mathrm{O}\mathrm{T}}=\mathcal{A}$.

For $N\ge k$ and $ϵ\in \{0,1{\}}^N$, put $R_{j,1}=P_j,R_{j,0}=I-P_j,Q_{ϵ}:={\prod }_{j=1}^N{R}_{j,{ϵ}_j}\mathrm{.}$ The $P_j$ commute, so the $Q_{ϵ}$ are pairwise orthogonal projections whose sum is $I$, and $S_N={\sum }_{ϵ\in \{0,1{\}}^N}\left({\sum }_{j=1}^N{3}^{-j}{ϵ}_j\right)Q_{ϵ}\mathrm{.}$ For every polynomial $p$, the finite orthogonal resolution gives $p(S_N)={\sum }_{ϵ}p({\sum }_{j=1}^N{3}^{-j}{ϵ}_j)Q_{ϵ}$. Uniform polynomial approximation and the isometric calculus [F8], [F10] pass this identity to $f_k$, so $f_k(S_N)={\sum }_{ϵ\in \{0,1{\}}^N}{f}_k\text{\hspace{0.17em}⁣}\left({\sum }_{j=1}^N{3}^{-j}{ϵ}_j\right)Q_{ϵ}=P_k,$ because the scalar sum belongs to $I_{({ϵ}_1,\dots ,{ϵ}_k)}$ and the function has value ${ϵ}_k$ there. The last equality uses ${\sum }_{ϵ:{ϵ}_k=1}{Q}_{ϵ}=P_k$.

The continuous calculus values converge in norm: $f_k(S_N)⟶f_k(S)\mathrm{.}$ For any $\delta >0$, [F10] gives a real polynomial $p$ with ${\sup }_{[-1\mathrm{/}2,1\mathrm{/}2]}\mathrm{\mid }{f}_k-p\mathrm{\mid }<\delta$. By [F7, F8], the spectra of $S_N$ and $S$ lie in this interval and the calculus is isometric, so each of $\mathrm{\parallel }{f}_k(S_N)-p(S_N)\mathrm{\parallel }$ and $\mathrm{\parallel }{f}_k(S)-p(S)\mathrm{\parallel }$ is below $\delta$. Since $S_N\to S$ in norm and multiplication is norm-continuous by [F13], $p(S_N)\to p(S)$; addition and scalar multiplication are norm-continuous by [F6]. Letting $\delta \downarrow 0$ proves the convergence.

Steps 8.2 and 8.3 show $P_k=f_k(S)$ for every $k$. The operator $S$ is self-adjoint and hence normal by [F14]; by [F8], $f_k(S)\in C^{\ast }(I,S)$, which is contained in the norm closure of the unital star-algebra generated by $S$ and therefore in $W^{\ast }(S)$. Hence $P_k\in W^{\ast }(S)$.

By [F5], $W^{\ast }(S)$ is the WOT closure of the unital star-algebra $D$ generated by $S$. This closure is itself a unital star-algebra: adjoint and each fixed left or right multiplication are WOT-continuous, as their matrix coefficients are WOT tests from [F2] and the adjoint identity [F12]. If $x,y\in {\bar{D}}^{\mathrm{W}\mathrm{O}\mathrm{T}}$, first approximate $y$ by elements of $D$ with left multiplication fixed at any $d\in D$ to get $dy\in {\bar{D}}^{\mathrm{W}\mathrm{O}\mathrm{T}}$; then approximate $x$ with right multiplication fixed at $y$ to get $xy\in {\bar{D}}^{\mathrm{W}\mathrm{O}\mathrm{T}}$. The adjoint map preserves the closure by the same continuity, and $I\in D$. Hence $W^{\ast }(S)$ is a WOT-closed unital star-algebra. It contains every $P_k$ by step 9.1, so it contains $\mathcal{C}$ and its WOT closure $\mathcal{A}$ by step 8.1. Conversely, $S\in \mathcal{A}$ by step 7.2 and $\mathcal{A}$ is a WOT-closed unital star-algebra, so the WOT closure of the unital star-algebra generated by $S$ is contained in $\mathcal{A}$. Thus $\mathcal{A}=W^{\ast }(S)$, with $S$ bounded and self-adjoint as required.