I have a question. We know by theory that any von Neumann algebra is direct integral of factors. Then how to get the decomposition in practical situation. Basically what is the decomposition examples for abelian vN algebras, Group vN algebras such that group is not i.c.c, and $\mathbb{B}(\mathcal{H})\otimes L^{\infty}(X,\mu)$. Thanks in advance!
On Direct integral decomposition of von Neumann algebras
c-star-algebrasfunctional-analysisvon-neumann-algebras
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The papers of Zimmer are good references, but they only cover the commutative case (the case of an action on a measure space). For actions of (locally compact) groups on general von Neumann algebras, amenability is defined and studied by Claire Anantharaman-Delaroche in the following papers:
Claire Anantharaman-Delaroche, Action moyennable d'un groupe localement compact sur une algèbre de von Neumann.(French. English summary)Math. Scand.45(1979), no.2, 289–304. https://www.mscand.dk/article/view/11844
Claire Anantharaman-Delaroche, Action moyennable d'un groupe localement compact sur une algèbre de von Neumann. II.(French. English summary)[Amenable action of a locally compact group on a von Neumann algebra. II] Math. Scand.50(1982), no.2, 251–268. https://www.mscand.dk/article/view/11958
In the first paper it is already proved that the crossed product $M\bar\rtimes_\alpha\Gamma$ if injective if and only if $M$ is injective and the action $\alpha$ is amenable. Here $\Gamma$ is a discrete group. For general locally compact groups only one direction of this holds.
For actions of (discrete) groups on $C^*$-algebras, amenability is defined and studied in the follow up paper:
Claire Anantharaman-Delaroche, Systèmes dynamiques non commutatifs et moyennabilité.(French)[Noncommutative dynamical systems and amenability] Math. Ann.279(1987), no.2, 297–315. https://link.springer.com/article/10.1007/BF01461725
In this paper it is proved that for an action $\alpha$ of a (discrete) group $\Gamma$ on a $C^*$-algebra $A$, the crossed product $A\rtimes_\alpha \Gamma$ is nuclear if and only if $A$ is nuclear and $\alpha$ is amenable.
Amenable actions of locally compact groups were only defined and studied recently in the preprint (still not published):
https://arxiv.org/abs/2003.03469 Amenability and weak containment for actions of locally compact groups on $C^*$-algebras, by Alcides Buss, Siegfried Echterhoff, Rufus Willett
Further references, and historical background, can be found in that preprint.
For $x\in G,$ $x\neq e,$ the conjugacy class of $x$ is the subset of $G$ defined by $$C_x=\{gxg^{-1}\:\, g\in G\}$$ The conjugacy classes do not change under the action of inner automorpisms. i.e. $hC_xh^{-1}=C_x$ for every $h\in G.$ According to Dietrich Burde comment
The von Neumann algebra $VN(G)$ of the group $G$ is a factor (the center of $VN(G)$ is trivial) if and only if for every $x\neq e$ the set $C_x$ is infinite.
For the proof of $\Rightarrow$ direction assume by contradiction that there is $x_0\in G,$ $x_0\neq e,$ such the set $C_{x_0}$ is finite. Consider the operator $$A=\sum_{x\in C_{x_0}}\lambda_x$$ Clearly $A$ belongs to the algebra generated by $\lambda_g,$ $g\in G,$ as well as to its strong operator closure, i.e. to $VN(G).$ The operator $A$ commutes with translations $\lambda_h$ for every $h\in G.$ Indeed $$\lambda_h A(\lambda_h)^{-1}=\lambda_h A\lambda_{h^{-1}}= \sum_{x\in C_{x_0}}\lambda_{hxh^{-1}}\underset{y=hxh^{-1}} {=}\sum_{y\in C_{x_0}}\lambda_y=A$$ The operator $A$ is nontrivial as $$\langle A\delta_e,\delta_{x_0}\rangle_{\ell^2(G)}=1$$ This completes the proof of $\Rightarrow$ direction.
For $\Leftarrow $ direction, assume that $VN(G)$ contains an operator $A,$ which commutes with all operators in $VN(G),$ hence it commutes with left translations $\lambda_g$ for every $g\in G.$ We are going to show that $A=0.$ Let $A\delta_e=\sum_{x\in G}a(x)\delta_x.$ The operator $A,$ restricted to the functions with finite support, is of the form $$A=\sum_{x\in G}a(x)\lambda_x$$ as it commutes with right translations $\rho_g.$ Indeed $$A\delta_y=A\rho_y(\delta_e)=\rho_y A\delta_e=\rho_y\left (\sum_{x\in G}a(x)\delta_x\right )=\sum_{x\in G} a(x)\delta_{xy}=\left (\sum_{x\in G} a(x)\lambda_x\right )\delta_y$$ As $A$ commutes with left translations we get $$\sum_{x\in G}a(x)\lambda_x=A=\lambda_{g^{-1}}A\lambda_g=\sum_{x\in G}a(x)\lambda_{g^{-1}xg}=\sum_{x\in G} a(gxg^{-1})\lambda_x$$ Therefore the function $a(x)$ is constant on each conjugacy class. Since $a(x)=A\delta_e\in \ell^2(G),$ the series $\sum_{x\in G} |a(x)|^2$ is convergent. As each conjugacy class is infinite then $a(x)\equiv 0,$ i.e $A=0.$
Best Answer
There is no "practical situation", my opinion (with a small caveat mentioned at the end). Any von Neumann algebra that is expressed to you in a somewhat concrete way, is more amenable to manipulation than a direct integral. People (very) seldom use direct integrals to prove some general fact, not to understand their algebras.
For example, an abelian von Neumann algebra is $L^\infty(X,\mu)$ for some measure space. Suppose $X$ is $\mathbb R^n$, or $\mathbb C^n$, or one of many many other nontrivial measure spaces. Writing your algebra as a direct integral of uncountable many copies of $\mathbb C$ gives you nothing. Similarly, on $B(H)\otimes L^\infty(X,\mu)$ you can see the centre directly (it's $I\otimes L^\infty(X,\mu)$) and do stuff; writing the algebra as a direct integral of uncountably many copies of $B(H)$ gives you nothing of value.
In the case of group von Neumann algebras I don't have much to say. Mostly because I don't really remember if one can characterize the centre explicitly: the only case where the direct integral decomposition can be useful is the case where you have minimal central projections, because in that case you can write the decomposition rather explicitly. That said, I don't think it is usual to be able to write projections in a group von Neumann algebras explicitly (I might be wrong, I haven't played with group algebras in a very long time).
The only case where the direct integral decomposition is meaningful is in the case of finite-dimensional algebras. A finite-dimensional von Neumann algebra is of the form $\bigoplus_{k=1}^m M_{m_k}(\mathbb C)$. That's precisely the direct integral decomposition (over a finite measure space, that's why it's tractable).