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, $\| \delta(u^k) \| \to \infty$.

[*] $N_0$ 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 $\Gamma$.

[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.

[Ch] Choda, Marie, Group factors of the Haagerup type, Proc. Japan Acad., Ser. A 59, 174-177 (1983). ZBL0523.46038.

[Pe] Peterson, Jesse, A 1-cohomology characterization of property (T) in von Neumann algebras, Pac. J. Math. 243, No. 1, 181-199 (2009). ZBL1178.22010.


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)

  • $\begingroup$ Thanks for the reference! Let me think a little bit whether this can be expressed in terms of unitaries before accepting it as an answer. $\endgroup$ – Adrián González-Pérez Mar 31 '20 at 18:43
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    $\begingroup$ Finding natural equivalent conditions to the Haagerup property based on the behavior of a derivation on unitaries might be difficult. This seems likely to be connected to the still open problem of whether the Haagerup property is equivalent to the compact approximation property. See Definition 4.13 from the paper: Anantharaman-Delaroche, Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. of Math. 171, no. 2, 309-341 (1995). $\endgroup$ – Jesse Peterson Apr 3 '20 at 15:51

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