## The $$(\ast\ast)$$-Haagerup property for $$C^\ast$$-algebras.(English)Zbl 1351.46055

In [Invent. Math. 50, 279–293 (1979; Zbl 0408.46046)], U. Haagerup introduced a property for discrete groups (later named after him) that generalizes amenability and that is satisfied by free groups. This property was subsequently transferred to von Neumann algebras and $$C^*$$-algebras. A von Neumann algebra $$M$$ with a normal, faithful, tracial state $$\tau$$ is said to have the Haagerup property if there exists a net $$(\Phi_j)_j$$ of normal, unital, completely positive maps $$\Phi_j: M\to M$$ such that: (1) for each $$j$$ one has $$\tau\circ\Phi_j\leq\tau$$ and $$\Phi_j$$ extends to a compact operator on $$L_2(M,\tau)$$; and (2) the net $$(\Phi_j)_j$$ converges pointwise to the identity in the $$2$$-norm induced by the trace $$\tau$$. This property was introduced by M. Choda [Proc. Japan Acad., Ser. A 59, 174–177 (1983; Zbl 0523.46038)], who also showed that a group has the Haagerup property if and only if so has its group von Neumann algebra (with the canonical trace).
A unital $$C^*$$-algebra $$A$$ with a faithful, tracial state $$\tau$$ is said to have the Haagerup property if there exists a net $$(\Phi_j)_j$$ of unital, completely positive maps $$\Phi_j: A\to A$$ satisfying the same conditions as in the von Neumann algebra case. (Thus, the only difference is that the maps $$\Phi_j$$ are not assumed to be normal.) This property was introduced by Z. Dong [J. Math. Anal. Appl. 377, No. 2, 631–644 (2011; Zbl 1218.46031)], who also showed that a group has the Haagerup property if and only if so has its reduced group $$C^*$$-algebra (with the canonical trace). The definition of the Haagerup property for von Neumann algebras does not depend on the choice of a faithful normal tracial state. Despite the definitions being analogous, this is not the case for $$C^*$$-algebras: It was shown by Y. Suzuki [J. Funct. Anal. 265, No. 8, 1778–1799 (2013; Zbl 1297.46040)] that there exists a $$C^*$$-algebra $$A$$ with faithful, tracial states $$\tau_1$$ and $$\tau_2$$ such that $$A$$ has the Haagerup property for $$\tau_1$$ but not for $$\tau_2$$.
The paper under review tries to resolve this problem by proposing a new definition of a Haagerup property for $$C^*$$-algebras, called the $$(**)$$-Haagerup property, that is independent of the choice of faithful tracial state. Following Definition 2.1, a unital $$C^*$$-algebra $$A$$ with faithful tracial state $$\tau$$ is said to have the $$(**)$$-Haagerup property if the bidual $$A^{**}$$ has the Haagerup property for von Neumann algebras.
As stated, the definition is of limited use since $$A^{**}$$ rarely has any faithful tracial state. For instance, if $$A$$ is a separable, non-type-$$\mathrm{I}$$ $$C^*$$-algebra (e.g., a unital, simple, infinite-dimensional $$C^*$$-algebra), then $$A$$ has type-$$\mathrm{III}$$ factor representations (e.g., by Proposition 6.8.6 on page 223 in G. K. Pedersen’s book [$$C^*$$-algebras and their automorphism groups. London-New York-San Francisco: Academic Press (1979; Zbl 0416.46043)]) and consequently $$A^{**}$$ does not have a faithful, tracial state.
The problem lies in Remark 2.1(2), where the authors claim that a faithful, tracial state $$\tau: A\to\mathbb{C}$$ induces a normal, faithful, tracial state $$\tau^{**}: A^{**}\to \mathbb{C}^{**}=\mathbb{C}$$. While $$\tau^{**}$$ is indeed a normal, tracial state, it is in general not faithful.
It is possible that the results of the paper can be obtained by considering a variant of the proposed $$(**)$$-Haagerup property where one requires that the type-$$\mathrm{II}_1$$ part of the bidual $$A^{**}$$ has the Haagerup property for von Neumann algebras.

### MSC:

 46L05 General theory of $$C^*$$-algebras 46L10 General theory of von Neumann algebras
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### References:

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