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**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.

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.

Reviewer: Hannes Thiel (Münster)

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\textit{C. Li} and \textit{X. Fang}, Chin. Ann. Math., Ser. B 37, No. 3, 367--372 (2016; Zbl 1351.46055)

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