zbMATH — the first resource for mathematics

Spectral gap characterization of full type III factors. (English) Zbl 1433.46040
Summary: We give a spectral gap characterization of fullness for type III factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if \(M\) is a full factor and \(\sigma:G\rightarrow\mathrm{Aut}(M)\) is an outer action of a discrete group \(G\) whose image in \(\mathrm{Out}(M)\) is discrete, then the crossed product von Neumann algebra \(M\rtimes_{\sigma}G\) is also a full factor. We apply this result to prove the following conjecture of Tomatsu-Ueda: the continuous core of a type \(\text{III}_1\) factor \(M\) is full if and only if \(M\) is full and its \(\tau\) invariant is the usual topology on \(\mathbb{R}\).

46L36 Classification of factors
46L40 Automorphisms of selfadjoint operator algebras
46L55 Noncommutative dynamical systems
Full Text: DOI arXiv
[1] H. Ando and U. Haagerup, Ultraproducts of von Neumann algebras, J. Funct. Anal. 266 (2014), no. 12, 6842-6913. · Zbl 1305.46049
[2] A. Connes, Almost periodic states and factors of type {III_{1}}, J. Funct. Anal. 16 (1974), no. 4, 415-445. · Zbl 0302.46050
[3] A. Connes, Classification of injective factors cases {II_{1}}, {II_{\infty}}, {III_{\lambda}}, {\lambda\neq 1}, Ann. of Math. (2) 104 (1976), 73-115. · Zbl 0343.46042
[4] A. Connes, Factors of type {III_{1}}, property {L^{\prime}_{\lambda}} and closure of inner automorphisms, J. Operator Theory 14 (1985), 189-211. · Zbl 0597.46063
[5] A. Connes and E. Størmer, Homogeneity of the state space of factors of type {III_{1}}, J. Funct. Anal. 28 (1978), no. 2, 187-196. · Zbl 0408.46048
[6] J. Dixmier and O. Maréchal, Vecteurs totalisateurs d’une algèbre de von Neumann, Comm. Math. Phys. 22 (1971), no. 1, 44-50. · Zbl 0211.44005
[7] U. Haagerup, {L^{p}}-spaces associated with an arbitrary von Neumann algebra, Algebres d’opérateurs et leurs applications en physique mathématique (Marseille 1977), Coll. Int. Centre Nat. Rech. Sci. 274, Editions du Centre National de la Recherche Scientifique, Paris (1979), 175-184.
[8] U. Haagerup, Operator valued weights in von Neumann algebras. I, J. Funct. Anal. 32 (1979), no. 2, 175-206. · Zbl 0426.46046
[9] V. F. Jones, Central sequences in crossed products of full factors, Duke Math. J 49 (1982), no. 1, 29-33. · Zbl 0492.46049
[10] H. Kosaki, Applications of uniform convexity of noncommutative {L^{p}}-spaces, Trans. Amer. Math. Soc. 283 (1984), no. 1, 265-282. · Zbl 0604.46064
[11] S. Popa, On the classification of inductive limits of {II_{1}} factors with spectral gap, Trans. Amer. Math. Soc. 364 (2012), no. 6, 2987-3000. · Zbl 1252.46063
[12] Y. Raynaud, On ultrapowers of non-commutative {L^{p}}-spaces, J. Operator Theory 48 (2002), no. 1, 41-68. · Zbl 1029.46102
[13] D. Shlyakhtenko, On the classification of full factors of type III, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4143-4159. · Zbl 1050.46046
[14] M. Takesaki, Operator algebras II, Springer, New York 2001.
[15] R. Tomatsu and Y. Ueda, A characterization of fullness of continuous cores of type {III_{1}} free product factors, Kyoto J. Math 56 (2014), no. 3, 599-610. · Zbl 1366.46049
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.