Faithfulness of bifree product states

Ramsey, Christopher

doi:10.1090/proc/14194

Faithfulness of bifree product states
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by Christopher Ramsey PDF

Proc. Amer. Math. Soc. 146 (2018), 5279-5288 Request permission

Abstract:

Given a nontrivial family of pairs of faces of unital $\mathrm {C}^*$-algebras where each pair has a faithful state, it is proved that if the bifree product state is faithful on the reduced bifree product of this family of pairs of faces, then each pair of faces arises as a minimal tensor product. A partial converse is also obtained.

References

Daniel Avitzour, Free products of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 271 (1982), no. 2, 423–435. MR 654842, DOI 10.1090/S0002-9947-1982-0654842-1
Kenneth J. Dykema, Faithfulness of free product states, J. Funct. Anal. 154 (1998), no. 2, 323–329. MR 1612705, DOI 10.1006/jfan.1997.3207
K. J. Dykema and M. Rørdam, Projections in free product $C^*$-algebras, Geom. Funct. Anal. 8 (1998), no. 1, 1–16. MR 1601917, DOI 10.1007/s000390050046
Amaury Freslon and Moritz Weber, On bi-free de Finetti theorems, Ann. Math. Blaise Pascal 23 (2016), no. 1, 21–51. MR 3505568, DOI 10.5802/ambp.353
Paul Skoufranis, On operator-valued bi-free distributions, Adv. Math. 303 (2016), 638–715. MR 3552536, DOI 10.1016/j.aim.2016.08.026
Masamichi Takesaki, On the cross-norm of the direct product of $C^{\ast }$-algebras, Tohoku Math. J. (2) 16 (1964), 111–122. MR 165384, DOI 10.2748/tmj/1178243737
Dan Voiculescu, Symmetries of some reduced free product $C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556–588. MR 799593, DOI 10.1007/BFb0074909
Dan-Virgil Voiculescu, Free probability for pairs of faces I, Comm. Math. Phys. 332 (2014), no. 3, 955–980. MR 3262618, DOI 10.1007/s00220-014-2060-7