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

Li, Changjing; Fang, Xiaochun

Chin. Ann. Math., Ser. B 37, No. 3, 367-372 (2016).

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.

Reviewer: Hannes Thiel (Münster)

MSC:

46L05	General theory of \(C^*\)-algebras
46L10	General theory of von Neumann algebras

Keywords:

\(C^\ast\)-algebras; von Neumann algebras; Haagerup property

Citations:

Zbl 0408.46046; Zbl 0523.46038; Zbl 1218.46031; Zbl 1297.46040; Zbl 0416.46043

References:

[1]	Blackadar, B., Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer-Verlag, New York, 2006. · Zbl 1092.46003
[2]	Brown, N. P. and Ozawa, N., C*-Algebras and Finite Dimensional Approximation, Grad. Stud. Math., Vol. 88, Amer. Math. Sci., Provindence, RI, 2008. · Zbl 1160.46001
[3]	Cherix, P. A., Cowling, M., Jolissaint, P., et al., The Haagerup Property for Groups, Gromov’s a-TMenability, Progr. Math., Vol. 197, Birkhauser, Basel, 2001. · Zbl 1030.43002
[4]	Choda, M., Group factors of the Haagerup type, Proc. Japan Acad., 59, 174-209, (1983) · Zbl 0523.46038
[5]	Connes, A.; Jonnes, V., Property T for von Neumann algebras, Bull. Lond. Math. Soc., 17, 57-62, (1985) · Zbl 1190.46047
[6]	Dong, Z., Haagerup property for C*-algebra, J. Math. Anal. Appl., 377, 631-644, (2011) · Zbl 1218.46031
[7]	Gromov, M., Asymptotic invariants of infinite groups, Geometic Group Theory, Vol. 2, G. A. Niblo and M. A. Roller (eds.), London Math. Soc. Lecture Notes, Vol. 182, Cambridge Univ. Press, Cambridge, 1993. · Zbl 0841.20039
[8]	Haagerup, U., An example of nonnuclear C*-algebra which has the metric approximation property, Invent. Math., 50, 279-293, (1979) · Zbl 0408.46046
[9]	Jolissaint, P., Haagerup approximation property for finite von Neumann algebra, J. Operator Theory, 48, 549-571, (2002) · Zbl 1029.46091
[10]	Pedersen, G., C*-Algebras and Their Automorphism Groups, Academic Press, London, 1979, 59-84.
[11]	Popa, S., On a class of type II1 factors with Betti numbers invariants, Ann. of Math., 163, 809-899, (2006) · Zbl 1120.46045
[12]	Ricard; Xu, Q., Khintchine type inequalities for reduced free products and applications, J. Reine Angew. Math., 599, 27-59, (2006) · Zbl 1170.46052
[13]	Suzuki, Y., Haagerup property for C*-algebra and rigidity of C*-algebra with property (T), J. Funct. Anal., 265, 1778-1799, (2013) · Zbl 1297.46040
[14]	Takesaki, M., Theory of Operator Algebras I, Springer-Verlag, New York, 1979, 120-130. · Zbl 0436.46043