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On graph products of multipliers and the Haagerup property for $C^{\ast }$ -dynamical systems



We consider the notion of the graph product of actions of discrete groups $\{G_{v}\}$ on a $C^{\ast }$ -algebra ${\mathcal{A}}$ and show that under suitable commutativity conditions the graph product action $\star _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6FC}_{v}:\star _{\unicode[STIX]{x1D6E4}}G_{v}\curvearrowright {\mathcal{A}}$ has the Haagerup property if each action $\unicode[STIX]{x1D6FC}_{v}:G_{v}\curvearrowright {\mathcal{A}}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.



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[1] Anantharaman-Delaroche, C.. Systèmes dynamiques non commutatifs et moyennabilité. Math. Ann. 279(2) (1987), 297315.
[2] Antolin, Y. and Dreesen, D.. The Haagerup property is stable under graph products. Preprint, 2013, arXiv:1305.6748.
[3] Atkinson, S.. Graph products of completely positive maps. J. Operator Theory 81(1) (2019), 133159.
[4] Baudisch, A.. Subgroups of semifree groups. Acta Math. Acad. Sci. Hungar. 38(1–4) (1981), 1928.
[5] Bédos, E. and Conti, R.. Negative definite functions for C -dynamical systems. Positivity 21(4) (2017), 16251646.
[6] Boca, F.. Free products of completely positive maps and spectral sets. J. Funct. Anal. 97(2) (1991), 251263.
[7] Brown, N. P. and Ozawa, N.. C -Algebras and Finite-dimensional Approximations (Graduate Studies in Mathematics, 88) . American Mathematical Society, Providence, RI, 2008.
[8] Caspers, M. and Fima, P.. Graph products of operator algebras. J. Noncommut. Geom. 11(1) (2017), 367411.
[9] Charney, R.. An introduction to right-angled Artin groups. Geom. Dedicata 125 (2007), 141158.
[10] Cherix, P.-A., Cowling, M., Jolissaint, P., Julg, P. and Valette, A.. Groups with the Haagerup Property (Modern Birkhäuser Classics) . Birkhäuser/Springer, Basel, 2001, Gromov’s a-T-menability, paperback reprint of the 2001 edition.
[11] Chiswell, I. M.. Right-angled Coxeter groups. Low-dimensional Topology and Kleinian Groups (Coventry/Durham, 1984) (London Mathematical Society Lecture Note Series, 112) . Cambridge University Press, Cambridge, 1986, pp. 297304.
[12] Choda, M.. Group factors of the Haagerup type. Proc. Japan Acad. Ser. A Math. Sci. 59(5) (1983), 174177.
[13] Combes, F.. Crossed products and Morita equivalence. Proc. Lond. Math. Soc. (3) 49(2) (1984), 289306.
[14] Connes, A. and Takesaki, M.. The flow of weights on factors of type III. Tôhoku Math. J. (2) 29(4) (1977), 473575.
[15] Dadarlat, M. and Guentner, E.. Constructions preserving Hilbert space uniform embeddability of discrete groups. Trans. Amer. Math. Soc. 355(8) (2003), 32533275.
[16] Dong, Z.. Haagerup property for C -algebras. J. Math. Anal. Appl. 377(2) (2011), 631644.
[17] Dong, Z. and Ruan, Z.-J.. A Hilbert module approach to the Haagerup property. Integral Equations Operator Theory 73(3) (2012), 431454.
[18] Droms, C.. Graph groups, coherence, and three-manifolds. J. Algebra 106(2) (1987), 484489.
[19] Droms, C.. Subgroups of graph groups. J. Algebra 110(2) (1987), 519522.
[20] Droms, C.. Isomorphisms of graph groups. Proc. Amer. Math. Soc. 100(3) (1987), 407408.
[21] Green, E. R.. Graph products of groups. PhD Thesis, University of Leeds, 1990.
[22] Haagerup, U.. An example of a nonnuclear C -algebra, which has the metric approximation property. Invent. Math. 50(3) (1978–1979), 279293.
[23] Nica, A. and Speicher, R.. Lectures on the Combinatorics of Free Probability (London Mathematical Society Lecture Note Series, 335) . Cambridge University Press, Cambridge, 2006.
[24] Popa, S.. Some rigidity results for non-commutative Bernoulli shifts. J. Funct. Anal. 230(2) (2006), 273328.
[25] Reckwerdt, E.. Weak amenability is stable under graph products. J. Lond. Math. Soc. (2) 96(1) (2017), 133155.
[26] Takesaki, M.. Theory of Operator Algebras I (Operator Algebras and Non-Commutative Geometry, 5, Encyclopaedia of Mathematical Sciences, 124) . Springer, Berlin, 2002, reprint of the first (1979) edition.
[27] Tu, J.-L.. La conjecture de Baum–Connes pour les feuilletages moyennables. K-Theory 17(3) (1999), 215264.
[28] Valette, A.. Weak amenability of right-angled Coxeter groups. Proc. Amer. Math. Soc. 119(4) (1993), 13311334.
[29] Wise, D. T.. From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry (CBMS Regional Conference Series in Mathematics, 117) . American Mathematical Society, Providence, RI, 2012, published for the Conference Board of the Mathematical Sciences, Washington, DC.


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