# zbMATH — the first resource for mathematics

Unique prime factorization for infinite tensor product factors. (English) Zbl 1418.46027
Summary: In this article, we investigate a unique prime factorization property for infinite tensor product factors. We provide several examples of type II and III factors which satisfy this property, including all free product factors with diffuse free product components. In the type III setting, this is the first classification result for infinite tensor product non-amenable factors. Our proof is based on Popa’s intertwining techniques and a characterization of relative amenability on the continuous cores.

##### MSC:
 46L36 Classification of factors
Full Text:
##### References:
 [1] Anantharaman-Delaroche, C., Amenable correspondences and approximation properties for von Neumann algebras, Pacific J. Math., 171, 309-341, (1995) · Zbl 0892.22004 [2] Ando, H.; Haagerup, U.; Houdayer, C.; Marrakchi, A., Structure of bicentralizer algebras and inclusions of type III factors, (2018), Preprint [3] Araki, H.; Woods, E. J., A classification of factors, Publ. RIMS Kyoto Univ. Ser. A, 3, 51-130, (1968) · Zbl 0206.12901 [4] Boutonnet, R.; Houdayer, C., Amenable absorption in amalgamated free product von Neumann algebras, Kyoto J. Math., (2018), in press · Zbl 1406.46045 [5] Boutonnet, R.; Houdayer, C.; Raum, S., Amalgamated free product type III factors with at most one Cartan subalgebras, Compos. Math., 150, 143-174, (2014) · Zbl 1308.46067 [6] Brown, N. P.; Ozawa, N., C^{⁎}-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics, vol. 88, (2008), American Mathematical Society: American Mathematical Society Providence, RI [7] Chifan, I.; Houdayer, C., Bass-Serre rigidity results in von Neumann algebras, Duke Math. J., 153, 23-54, (2010) · Zbl 1201.46057 [8] Connes, A., Une classification des facteurs de type III, Ann. Sci. Éc. Norm. Supér. (4), 6, 133-252, (1973) · Zbl 0274.46050 [9] Connes, A., Classification of injective factors, Ann. of Math. (2), 104, 73-115, (1976) · Zbl 0343.46042 [10] Connes, A.; Størmer, E., Homogeneity of the state space of factors of type $$\operatorname{II} \operatorname{I}_1$$, J. Funct. Anal., 28, 2, 187-196, (1978) · Zbl 0408.46048 [11] Ge, L., Applications of free entropy to finite von Neumann algebras, II, Ann. of Math., 147, 143-157, (1998) · Zbl 0924.46050 [12] Haagerup, U., Operator valued weights in von Neumann algebras, I, J. Funct. Anal., 32, 175-206, (1979) · Zbl 0426.46046 [13] Haagerup, U., Operator valued weights in von Neumann algebras, II, J. Funct. Anal., 33, 339-361, (1979) · Zbl 0426.46047 [14] Haagerup, U., Connes’ bicentralizer problem and uniqueness of the injective factor of type $$\operatorname{II} \operatorname{I}_1$$, Acta Math., 158, 1-2, 95-148, (1987) · Zbl 0628.46061 [15] Houdayer, C.; Isono, Y., Unique prime factorization and bicentralizer problem for a class of type III factors, Adv. Math., 305, 402-455, (2017) · Zbl 1371.46050 [16] Houdayer, C.; Ricard, E., Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors, Adv. Math., 228, 764-802, (2011) · Zbl 1267.46071 [17] Houdayer, C.; Ueda, Y., Asymptotic structure of free product von Neumann algebras, Math. Proc. Cambridge Philos. Soc., 161, 489-516, (2016) · Zbl 1379.46047 [18] Houdayer, C.; Ueda, Y., Rigidity of free product von Neumann algebras, Compos. Math., 152, 2461-2492, (2016) · Zbl 1379.46046 [19] Ioana, A., Cartan subalgebras of amalgamated free product $$\operatorname{I} \operatorname{I}_1$$ factors (with an appendix joint with S. Vaes), Ann. Sci. Éc. Norm. Supér., 48, 71-130, (2015) [20] Ioana, A.; Peterson, J.; Popa, S., Amalgamated free products of weakly rigid factors and calculation of their symmetry groups, Acta Math., 200, 85-153, (2008) · Zbl 1149.46047 [21] Isono, Y., Weak exactness for $$C^\ast$$-algebras and application to condition (AO), J. Funct. Anal., 264, 964-998, (2013) · Zbl 1284.46046 [22] Isono, Y., Examples of factors which have no Cartan subalgebras, Trans. Amer. Math. Soc., 367, 7917-7937, (2015) · Zbl 1342.46056 [23] Isono, Y., On bi-exactness of discrete quantum groups, Int. Math. Res. Not. IMRN, 11, 3619-3650, (2015) · Zbl 1332.46072 [24] Isono, Y., Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors, (2016), Preprint [25] Isono, Y., On fundamental groups of tensor product $$\operatorname{I} \operatorname{I}_1$$ factors, (2016), Preprint [26] Isono, Y., Some prime factorization results for free quantum group factors, J. Reine Angew. Math., 722, 215-250, (2017) · Zbl 1445.46044 [27] Krieger, W., On ergodic flows and the isomorphism of factors, Math. Ann., 223, 19-70, (1976) · Zbl 0332.46045 [28] Ozawa, N., Solid von Neumann algebras, Acta Math., 192, 111-117, (2004) · Zbl 1072.46040 [29] Ozawa, N.; Popa, S., Some prime factorization results for type $$\operatorname{I} \operatorname{I}_1$$ factors, Invent. Math., 156, 223-234, (2004) · Zbl 1060.46044 [30] Ozawa, N.; Popa, S., On a class of $$\operatorname{I} \operatorname{I}_1$$ factors with at most one Cartan subalgebra, Ann. of Math. (2), 172, 713-749, (2010) · Zbl 1201.46054 [31] Peterson, J., L^{2}-rigidity in von Neumann algebras, Invent. Math., 175, 2, 417-433, (2009) · Zbl 1170.46053 [32] Popa, S., On a problem of R. V. Kadison on maximal abelian ⁎-subalgebras in factors, Invent. Math., 65, 2, 269-281, (1981) · Zbl 0481.46028 [33] Popa, S., Orthogonal pairs of ⁎-subalgebras in finite von Neumann algebras, J. Operator Theory, 9, 2, 253-268, (1983) · Zbl 0521.46048 [34] Popa, S., On a class of type $$\operatorname{I} \operatorname{I}_1$$ factors with Betti numbers invariants, Ann. of Math., 163, 809-899, (2006) · Zbl 1120.46045 [35] Popa, S., Strong rigidity of $$\operatorname{I} \operatorname{I}_1$$ factors arising from malleable actions of w-rigid groups I, Invent. Math., 165, 369-408, (2006) · Zbl 1120.46043 [36] Popa, S.; Vaes, S., Unique Cartan decomposition for $$\operatorname{I} \operatorname{I}_1$$ factors arising from arbitrary actions of free groups, Acta Math., 212, 141-198, (2014) · Zbl 1307.46047 [37] Powers, R. T., Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math., 86, 138-171, (1967) · Zbl 0157.20605 [38] Sauvageot, J., Sur le produit tensoriel relatif d’espaces de Hilbert, J. Operator Theory, 9, 2, 237-252, (1983) · Zbl 0517.46050 [39] Takesaki, M., Theory of Operator Algebras II, Operator Algebras and Non-Commutative Geometry, vol. 5, (2002), Springer-Verlag: Springer-Verlag Berlin, Encyclopedia of Mathematical Sciences, vol. 125 · Zbl 0990.46034 [40] Ueda, Y., Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math., 228, 2647-2671, (2011) · Zbl 1252.46059
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.