## Strong Singularity For Subalgebras Of Finite Factors

Venue: | Internat. J. Math |

Citations: | 23 - 6 self |

### BibTeX

@ARTICLE{Robertson_strongsingularity,

author = {Guyan Robertson and Allan Sinclair and Roger R. Smith},

title = {Strong Singularity For Subalgebras Of Finite Factors},

journal = {Internat. J. Math},

year = {},

volume = {14},

pages = {235--258}

}

### OpenURL

### Abstract

In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type II 1 factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity. 2000 Mathematics Subject Classification Numbers: 46L10, 22D25 Partially supported by the Australian Research Council Partially supported by the National Science Foundation. 1

### Citations

373 | Metric Spaces of Non-positive Curvature - Bridson, Haefliger - 1999 |

356 |
Fundamentals of the Theory of Operator Algebras
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(Show Context)
Citation Context ...generated by the unitary which implements the action. When the action is ergodic and the automorphisms are trace preserving, it is well known that the resulting crossed product is a type II 1 factor, =-=[11]-=-. Under the additional hypothesis that the action is either strongly or weakly mixing (Lemmas 3.1 and 3.2), we obtain strong singularity of B, showing that such masas arise naturally from classical er... |

191 |
Ergodic Theory
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Citation Context ...x j # i (y k,r ) - tr(x j )tr(y k,r )| # # (3.12) for all n # 1, and this violates the defining inequality (3.4) of weakly mixing. This completes the proof . Remark 3.3. Classical ergodic theory (see =-=[15]-=-) provides many examples of strongly mixing transformations of measure spaces, as well as examples which are weakly but not strongly mixing. The two previous lemmas then give examples of strongly sing... |

189 |
The rigidity of locally symmetric spaces
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(Show Context)
Citation Context ...ntrary to hypothesis. The result now follows from Lemma 4.1. In the first class of examples, # is the fundamental group of a compact locally symmetric space of nonpositive curvature. The classic book =-=[13]-=- is a convenient reference for the background and necessary results. There is a clear introduction to the theory of symmetric spaces in [2, Chapter II.10]. Let X be a symmetric space of noncompact typ... |

161 | Manifolds of nonpositive curvature - Ballmann, Gromov, et al. - 1985 |

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Groupes réductifs sur un corps local
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(Show Context)
Citation Context ...s particularly evident if one considers groups of p-adic type. Specifically, let G be a connected semisimple group defined over a nonarchimedean local field. Then G acts on its Bruhat-Tits building # =-=[4]-=-, and the vertex set of # may be identified with G/K, where K is a maximal compact subgroup. We refer to [18] for the general theory of buildings. It is worth making a few remarks about the structure ... |

91 | la Harpe, Topics in geometric group theory - de - 2000 |

42 | Small cancellation theory and automatic groups - Gersten, Short - 1991 |

33 |
Groups acting simply transitively on the vertices of a building of type
- Cartwright, Mantero, et al.
(Show Context)
Citation Context ...tivating examples here come from [17], which was our starting point in constructing strongly singular masas. The building examples are of particular interest because many of the groups constructed in =-=[5]-=- do not embed naturally into linear groups. In the final section of the paper, we give another class of examples based on, but extending, those of Dixmier, [6]. As before, the geometry of the spaces o... |

26 | Strongly singular masas in type II1 factors, preprint
- Sinclair, Smith
- 2001
(Show Context)
Citation Context ...g from discrete groups. However, it is a difficult problem to decide whether a given masa is singular, and this prompted the second and third authors to introduce the concept of strong singularity in =-=[20]-=-. The trace induces a norm ‖ · ‖2 on M by ‖x‖2 = (tr(x∗x))1/2, and a norm ‖ · ‖∞,2 may then be defined for a map φ :M→M by ‖φ‖∞,2 = sup{‖φ(x)‖2 : ‖x‖ ≤ 1}. (1.2) Letting EN denote the unique trace pre... |

23 |
Singular maximal abelian ∗-subalgebras in continuous von Neumann algebras
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(Show Context)
Citation Context ...nal section of the paper, we give another class of examples based on, but extending, those of Dixmier, [6]. As before, the geometry of the spaces on which our groups act is the crucial ingredient. In =-=[16]-=-, Popa was able to construct singular masas in any type II 1 factor. At this time, we do not know if this is also true for strongly singular masas, or indeed whether all singular masas must also be st... |

20 | Compact manifolds of nonpositive curvature - Lawson, Yau - 1972 |

18 |
Actions of Cartan Subgroups
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(Show Context)
Citation Context ...mute and generate a free abelian subgroup of rank two inside #, satisfying the hypotheses of Theorem 5.8. Example 5.12. [Groups acting on products of trees] Consider some specific examples studied in =-=[14]-=-. In [14, Section 3], there is constructed a lattice subgroup # of G = PGL 2 (Q p ) PGL 2 (Q l ), where p, l # 1 (mod 4) are two distinct primes. This restriction is made because -1 has a square root ... |

2 |
Type analysis of the regular representation of a nonunimodular group
- Sutherland
- 1978
(Show Context)
Citation Context ...t product # = Ho # K is the set of formal products {hk : h # H, k # K} with multiplication (hk)(h # k # ) = (h# k (h # ))(kk # ). (2.7) The action # lifts from H to V N(H), and V N(#) = V N(H) o # K, =-=[21]-=-. We assume this notation in the next result. Identity elements of groups are denoted e H or e K , and the abbreviation I.C.C. means infinite conjugacy class. Theorem 2.2. Let H and K be infinite disc... |

2 |
Maximal subalgebras of the group factor of an Ã2
- Robertson, Steger
- 1996
(Show Context)
Citation Context ...fundamental group of a compact locally symmetric space of nonpositive curvature, while in Section 5, Γ acts by isometries on locally finite euclidean buildings. The motivating examples here come from =-=[17]-=-, which was our starting point in constructing strongly singular masas. The building examples are of particular interest because many of the groups constructed in [5] do not embed naturally into linea... |

1 |
Sous-anneaux abeliens maximaux dans les facteurs de type
- Diximer
- 1954
(Show Context)
Citation Context ...ralian Research Council (2) Partially supported by the National Science Foundation. 1 Introduction Let A be a maximal abelian self--adjoint subalgebra (masa) in a type II 1 factor M with trace tr. In =-=[6]-=-, Dixmier identified various classes of masas based on the structure of the normalizer N(A) = {u # M : uAu # = A, u unitary}. (1.1) In particular, A is said to be singular if N(A) # A, so that the onl... |

1 |
Maximal subalgebras of the group factor of an A 2
- Robertson, Steger
- 1996
(Show Context)
Citation Context ...fundamental group of a compact locally symmetric space of nonpositive curvature, while in Section 5, # acts by isometries on locally finite euclidean buildings. The motivating examples here come from =-=[17]-=-, which was our starting point in constructing strongly singular masas. The building examples are of particular interest because many of the groups constructed in [5] do not embed naturally into linea... |

1 |
Strongly singular masas
- Sinclair, Smith
- 2002
(Show Context)
Citation Context ...ing from discrete groups. However, it is a di#cult problem to decide whether a given masa is singular, and this prompted the second and third authors to introduce the concept of strong singularity in =-=[20]-=-. The trace induces a norm #s# 2 on M by #x# 2 = (tr(x # x)) 1/2 , and a norm #s# #,2 may then be defined for a map # : M#M by ####,2 = sup{##(x)# 2 : #x# # 1}. (1.2) Letting E N denote the unique tra... |