A correspondence $\_{N} H\_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras was proved in [Pe]. 



> 
> **Theorem ([Pe,Theorem 3.2.])** Let $N$ be a factor and $N\_0 \subset N$ weakly-$\ast$ dense, unital $\ast$-subalgebra containing a
>  non-$\Gamma$-set\*. Then, the following are equivalent:
> 
> 
> 1. $N$ has property (T).
> 2. Let $H = \_{N}H\_N$ be any correspondence. Any *a priori* unbounded closable derivation $\delta: D(\delta) \subset N \to H$ defined over a
>  domain $N\_0 \subset D(\delta)$ is inner.
> 
> 
> 


The intuition in the theorem above is that derivations $\delta$ over a correspondence are the 
von Neumann analogue of $1$-cocycles over a unitary representation. The characterization in point 2 is thus the analogue of the characterization of property (T) groups as those whose $1$-cocyles are all inner, which is due to Delorme/Guichardet.


There is a notion of Haagerup property for factors that was introduced in [Ch].



> 
> **Question:** Is there a characterization of the Haagerup approximation property for finite von Neumann algebras in terms of unbounded derivations?
> 
> 
> 


A group has the Haagerup approximation property iff it admits a metrically proper $1$-cocycle $\beta: G \to H$, ie a cocycle such that $\| \beta(g\_n) \| \to \infty$ as $g\_n$ escapes from every compact subset of $G$. What is the analogue of properness for cocycles? 


A characterization of Haagerup property for von Neumann algebras in terms of quantum Markov semigroups has appeared in [JM]. I guess the natural thing to do would be to use the theory developed by Cipriani/Sauvageot to obtain a derivation from the Markovian semigroup and try to characterize which derivations correspond to semigroups of compact operators. Is this characterization known already in the literature?


A guess would be those derivations such that, for every unitary $u \in U(N\_0)$ such that $u^k \to 0$ weakly, $\| \delta(u^k) \| \to \infty$.




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[\*] $N\_0$ contains a non-$\Gamma$ set if there is a finite $F \subset N\_0$ and a $K > 0$ such that for all $\xi \in L^2(N , \tau )$, we have $\| \xi - \tau(\xi) \|\_2^2 \leq K \, \sum\_{x \in F} \|[\xi,x]\|\_2^2$, i.e.: a set implementing the failure of property $\Gamma$.




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[JM] *Jolissaint, Paul; Martin, Florian*, Finite von Neumann algebras with Haagerup property and (L^2)-compact semigroups, Bull. Belg. Math. Soc. - Simon Stevin 11, No. 1, 35-48 (2004). [ZBL1068.46037](https://zbmath.org/?q=an:1068.46037).


[Ch] *Choda, Marie*, [**Group factors of the Haagerup type**](http://dx.doi.org/10.3792/pjaa.59.174), Proc. Japan Acad., Ser. A 59, 174-177 (1983). [ZBL0523.46038](https://zbmath.org/?q=an:0523.46038).


[Pe] *Peterson, Jesse*, [**A 1-cohomology characterization of property (T) in von Neumann algebras**](http://dx.doi.org/10.2140/pjm.2009.243.181), Pac. J. Math. 243, No. 1, 181-199 (2009). [ZBL1178.22010](https://zbmath.org/?q=an:1178.22010).



Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations
For separable von Neumann algebras the Haagerup property for $N$ is equivalent to the existence of a real closable derivation $\delta$ such that $\delta^\*\delta$ has compact resolvents in $\mathcal B(L^2(N))$. This is true even in the non-tracial case. See Theorem 7.7 in: 


Martijn Caspers, Adam Skalski, The Haagerup Approximation Property for von Neumann Algebras via Quantum Markov Semigroups and Dirichlet Forms, Commun. Math. Phys. 336, 1637–1664 (2015)


