×

zbMATH — the first resource for mathematics

Unique prime factorization and bicentralizer problem for a class of type III factors. (English) Zbl 1371.46050
Summary: We show that whenever \(m \geq 1\) and \(M_1, \dots, M_m\) are nonamenable factors in a large class of von Neumann algebras that we call \(\mathcal{C}_{(\text{AO})}\) and which contains all free Araki-Woods factors, the tensor product factor \(M_1 \overline{\otimes} \cdots \overline{\otimes} M_m\) retains the integer \(m\) and each factor \(M_i\) up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from [N. Ozawa and S. Popa, Invent. Math. 156, No. 2, 223–234 (2004; Zbl 1060.46044); Y. Isono, J. Reine Angew. Math. 722, 215–250 (2017; Zbl 1445.46044)] and moreover provides new UPF results in the case when \(M_1, \dots, M_m\) are free Araki-Woods factors. In order to obtain the aforementioned UPF results, we show that Connes’s bicentralizer problem has a positive solution for all type \(\operatorname{III}_1\) factors in the class \(\mathcal{C}_{(\mathrm{AO})}\).

MSC:
46L36 Classification of factors
46L10 General theory of von Neumann algebras
46L06 Tensor products of \(C^*\)-algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ando, H.; Haagerup, U., Ultraproducts of von Neumann algebras, J. Funct. Anal., 266, 6842-6913, (2014) · Zbl 1305.46049
[2] Blackadar, B., Operator algebras, Encyclopaedia of Mathematical SciencesOperator Algebras and Non-commutative Geometry, 3, vol. 122, (2006), Springer-Verlag Berlin, xx+517 pp
[3] Brown, N. P.; Ozawa, N., C^{⁎}-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, (2008), American Mathematical Society Providence, RI, xvi+509 pp · Zbl 1160.46001
[4] Chifan, I.; Sinclair, T.; Udrea, B., On the structural theory of \(\text{II}_1\) factors of negatively curved groups, II. actions by product groups, Adv. Math., 245, 208-236, (2013) · Zbl 1288.46037
[5] Chifan, I.; Kida, Y.; Pant, S., Primeness results for von Neumann algebras associated with surface braid groups, Int. Math. Res. Not., (2015), 42 pp
[6] Choi, M. D.; Effros, E. G., The completely positive lifting problem for \(C^\ast\)-algebras, Ann. of Math., 104, 585-609, (1976) · Zbl 0361.46067
[7] Connes, A., Une classification des facteurs de type III, Ann. Sci. Éc. Norm. Supér., 6, 133-252, (1973) · Zbl 0274.46050
[8] Connes, A., Almost periodic states and factors of type \(\operatorname{I} \operatorname{I} \operatorname{I}_1\), J. Funct. Anal., 16, 415-445, (1974) · Zbl 0302.46050
[9] Connes, A., Classification of injective factors. cases \(\operatorname{I} \operatorname{I}_1\), \(\operatorname{I} \operatorname{I}_\infty\), \(\operatorname{I} \operatorname{I} \operatorname{I}_\lambda\), \(\lambda \ne 1\), Ann. of Math., 74, 73-115, (1976)
[10] Connes, A.; Takesaki, M., The flow of weights of factors of type III, Tôhoku Math. J., 29, 473-575, (1977) · Zbl 0408.46047
[11] Cuntz, J., Simple C^{⁎}-algebras generated by isometries, Comm. Math. Phys., 57, 173-185, (1977) · Zbl 0399.46045
[12] Gao, M.; Junge, M., Examples of prime von Neumann algebras, Int. Math. Res. Not. IMRN, 15, (2007) · Zbl 1134.46038
[13] Haagerup, U., The standard form of von Neumann algebras, Math. Scand., 37, 271-283, (1975) · Zbl 0304.46044
[14] Haagerup, U., Operator valued weights in von Neumann algebras, I, J. Funct. Anal., 32, 175-206, (1979) · Zbl 0426.46046
[15] Haagerup, U., Operator valued weights in von Neumann algebras, II, J. Funct. Anal., 33, 339-361, (1979) · Zbl 0426.46047
[16] Haagerup, U., Connes’ bicentralizer problem and uniqueness of the injective factor of type \(\operatorname{I} \operatorname{I} \operatorname{I}_1\), Acta Math., 69, 95-148, (1986)
[17] Houdayer, C., Sur la classification de certaines algèbres de von Neumann, (2007), Université Paris VII, PhD thesis
[18] Houdayer, C., Free Araki-Woods factors and Connes’ bicentralizer problem, Proc. Amer. Math. Soc., 137, 3749-3755, (2009) · Zbl 1183.46060
[19] Houdayer, C., Gamma stability in free product von Neumann algebras, Comm. Math. Phys., 336, 831-851, (2015) · Zbl 1328.46046
[20] Houdayer, C.; Raum, S., Asymptotic structure of free Araki-Woods factors, Math. Ann., 363, 237-267, (2015) · Zbl 1339.46057
[21] Houdayer, C.; Ueda, Y., Rigidity of free product von Neumann algebras, Compos. Math., (2016), in press · Zbl 1379.46046
[22] Houdayer, C.; Vaes, S., Type III factors with unique Cartan decomposition, J. Math. Pures Appl., 100, 564-590, (2013) · Zbl 1291.46052
[23] Isono, Y., Examples of factors which have no Cartan subalgebras, Trans. Amer. Math. Soc., 367, 7917-7937, (2015) · Zbl 1342.46056
[24] Isono, Y., On bi-exactness of discrete quantum groups, Int. Math. Res. Not. IMRN, 11, 3619-3650, (2015) · Zbl 1332.46072
[25] Isono, Y., Some prime factorization results for free quantum group factors, J. Reine Angew. Math., (2016), in press
[26] Izumi, M.; Longo, R.; Popa, S., A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal., 155, 25-63, (1998) · Zbl 0915.46051
[27] Kadison, R. V.; Ringrose, J. R., Fundamentals of the theory of operator algebras. vol. II. advanced theory, Graduate Studies in Mathematics, vol. 16, i-xxii, (1997), American Mathematical Society Providence, RI, and 399-1074
[28] Kosaki, H., Extension of Jones theory on index to arbitrary factors, J. Funct. Anal., 66, 123-140, (1986) · Zbl 0607.46034
[29] Kosaki, H., Characterization of crossed product (properly infinite case), Pacific J. Math., 137, 159-167, (1989) · Zbl 0693.46058
[30] Ocneanu, A., Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, vol. 1138, (1985), Springer-Verlag Berlin, iv+115 pp · Zbl 0608.46035
[31] Ozawa, N., Solid von Neumann algebras, Acta Math., 192, 111-117, (2004) · Zbl 1072.46040
[32] Ozawa, N., A kurosh type theorem for type \(\text{II}_1\) factors, Int. Math. Res. Not., (2006), 21 pp
[33] 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
[34] Peterson, J., \(\text{L}^2\)-rigidity in von Neumann algebras, Invent. Math., 175, 417-433, (2009) · Zbl 1170.46053
[35] Pimsner, M.; Popa, S., Entropy and index for subfactors, Ann. Sci. Éc. Norm. Supér., 19, 57-106, (1986) · Zbl 0646.46057
[36] Popa, S., On a problem of R.V. kadison on maximal abelian ⁎-subalgebras in factors, Invent. Math., 65, 269-281, (1981) · Zbl 0481.46028
[37] Popa, S., On a class of type \(\text{II}_1\) factors with Betti numbers invariants, Ann. of Math., 163, 809-899, (2006) · Zbl 1120.46045
[38] Popa, S., Strong rigidity of \(\text{II}_1\) factors arising from malleable actions of w-rigid groups I, Invent. Math., 165, 369-408, (2006) · Zbl 1120.46043
[39] Sako, H., Measure equivalence rigidity and bi-exactness of groups, J. Funct. Anal., 257, 10, 3167-3202, (2009) · Zbl 1256.37002
[40] Shlyakhtenko, D., Free quasi-free states, Pacific J. Math., 177, 329-368, (1997) · Zbl 0882.46026
[41] Sizemore, J. O.; Winchester, A., A unique prime decomposition result for wreath product factors, Pacific J. Math., 265, 221-232, (2013) · Zbl 1331.46048
[42] Takesaki, M., Theory of operator algebras I, Encyclopedia of Mathematical SciencesOperator Algebras and Non-commutative Geometry, 5, vol. 124, (2002), Springer-Verlag Berlin, xix+415 pp · Zbl 0990.46034
[43] Takesaki, M., Theory of operator algebras. II, Encyclopaedia of Mathematical SciencesOperator Algebras and Non-commutative Geometry, 6, vol. 125, (2003), Springer-Verlag Berlin, xxii+518 pp · Zbl 1059.46031
[44] Ueda, Y., Some analysis on amalgamated free products of von Neumann algebras in non-tracial setting, J. Lond. Math. Soc., 88, 25-48, (2013) · Zbl 1285.46048
[45] Vaes, S., États quasi-libres libres et facteurs de type III (d’après D. shlyakhtenko), Séminaire Bourbaki, exposé 937, Astérisque, 299, 329-350, (2005) · Zbl 1091.46037
[46] Vaes, S., Explicit computations of all finite index bimodules for a family of \(\text{II}_1\) factors, Ann. Sci. Éc. Norm. Supér., 41, 743-788, (2008) · Zbl 1194.46086
[47] Vaes, S.; Vander Vennet, N., Poisson boundary of the discrete quantum group \(\hat{A_u(F)}\), Compos. Math., 146, 1073-1095, (2010) · Zbl 1204.46035
[48] Vaes, S.; Vergnioux, R., The boundary of universal discrete quantum groups, exactness, and factoriality, Duke Math. J., 140, 35-84, (2007) · Zbl 1129.46062
[49] Voiculescu, D.-V.; Dykema, K. J.; Nica, A., Free random variables, CRM Monograph Series, vol. 1, (1992), American Mathematical Society Providence, RI · Zbl 0795.46049
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.