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

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: