×

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
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.