Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

Question

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.

[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.

Answer 1

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)