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23
Strong singularity of singular masas in II1 factors
 Illinois J. Math
"... Abstract. A singular masa A in a II1 factor N is defined by the property that any unitary w ∈ N for which A = wAw ∗ must lie in A. A strongly singular masa A is one that satisfies the inequality ‖EA − EwAw ∗ ‖∞,2 ≥ ‖w − EA(w)‖2 for all unitaries w ∈ N, where EA is the conditional expectation of N on ..."
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Cited by 10 (4 self)
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Abstract. A singular masa A in a II1 factor N is defined by the property that any unitary w ∈ N for which A = wAw ∗ must lie in A. A strongly singular masa A is one that satisfies the inequality ‖EA − EwAw ∗ ‖∞,2 ≥ ‖w − EA(w)‖2 for all unitaries w ∈ N, where EA is the conditional expectation of N onto A, and ‖ · ‖∞,2 is defined for bounded maps φ: N → N by sup{‖φ(x)‖2: x ∈ N, ‖x ‖ ≤ 1}. Strong singularity easily implies singularity, and the main result of this paper shows the reverse implication. 1.
VALUES OF THE PUKÁNSZKY INVARIANT IN FREE GROUP FACTORS AND THE HYPERFINITE FACTOR
, 2005
"... Let A ⊆ M ⊆ B(L 2 (M)) be a maximal abelian selfadjoint subalgebra (masa) in a type II1 factor M in its standard representation. The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection eA ∈ A onto L 2 (A). Then A ′ (1 −eA) decomposes into a dire ..."
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Cited by 5 (1 self)
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Let A ⊆ M ⊆ B(L 2 (M)) be a maximal abelian selfadjoint subalgebra (masa) in a type II1 factor M in its standard representation. The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection eA ∈ A onto L 2 (A). Then A ′ (1 −eA) decomposes into a direct sum of type In algebras for n ∈ {1,2, · · · , ∞}, and those n’s which occur in the direct sum form a set called the Pukánszky invariant, Puk(A), also denoted PukM(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S ∪{∞} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of N. The only previously known values for masas in free group factors were {∞} and {1, ∞}, and some values of the form S ∪ {∞} are new also for the hyperfinite factor. We also consider a more refined invariant (that we will call the measure–multiplicity invariant), which was considered recently by Neshveyev and Størmer and has been known to experts for a long time. We use the measure–multiplicity invariant to distinguish two masas in a free group factor, both having Pukánszky invariant {n, ∞}, for arbitrary n ∈ N.
Normalizers of irreducible subfactors
, 2007
"... We consider normalizers of an irreducible inclusion N ⊆ M of II1 factors. In the infinite index setting an inclusion uNu ∗ ⊆ N can be strict, forcing us to also investigate the semigroup of onesided normalizers. We relate these normalizers of N in M to projections in the basic construction and sho ..."
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Cited by 4 (4 self)
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We consider normalizers of an irreducible inclusion N ⊆ M of II1 factors. In the infinite index setting an inclusion uNu ∗ ⊆ N can be strict, forcing us to also investigate the semigroup of onesided normalizers. We relate these normalizers of N in M to projections in the basic construction and show that every trace one projection in the relative commutant N ′ ∩ 〈M,eN 〉 is of the form u ∗ eNu for some unitary u ∈ M with uNu ∗ ⊆ N. This enables us to identify the normalizers and the algebras they generate in several situations. In particular each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of irreducible subfactors arising from subgroup–group inclusions H ⊆ G. Here the normalizers are the normalizing group elements modulo a unitary from L(H). We are also able to identify the finite trace L(H)bimodules in ℓ2 (G) as double cosets which are also finite unions of left cosets. In this paper we consider the following general problem: given an irreducible inclusion
Strongly singular MASA’s and mixing actions in finite von Neumann algebras
, 2008
"... Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A = L(Γ0) is a singular MASA in M = L(Γ) which satisfies a weakly mixing condition. If moreover it satisfies a ..."
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Cited by 3 (0 self)
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Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A = L(Γ0) is a singular MASA in M = L(Γ) which satisfies a weakly mixing condition. If moreover it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, HNN extentions and semidirect products, and in particular we exhibit examples of singular MASA’s that satisfy the weak mixing condition but not the strong mixing one.
Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces
 ArXiv:math.OA/0504362 v1
, 2005
"... Abstract. Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra VN(Γ) is the convolution algebra of functions f ∈ ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let T r be a totally geodesic flat torus of dimension r in M and let Γ0 ∼ = Z r be the ..."
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Cited by 3 (1 self)
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Abstract. Consider a compact locally symmetric space M of rank r, with fundamental group Γ. The von Neumann algebra VN(Γ) is the convolution algebra of functions f ∈ ℓ2(Γ) which act by left convolution on ℓ2(Γ). Let T r be a totally geodesic flat torus of dimension r in M and let Γ0 ∼ = Z r be the image of the fundamental group of T r in Γ. Then VN(Γ0) is a maximal abelian ⋆subalgebra of VN(Γ) and its unitary normalizer is as small as possible. If M has constant negative curvature then the Pukánszky invariant of VN(Γ0) is ∞. 1.
On completely singular von Neumann subalgebras
 Proc. Edinb. Math. Soc
"... Let M be a von Neumann algebra acting on a Hilbert space H, and N be a singular von Neumann subalgebra of M. If N ¯ ⊗ B(K) is singular in M ¯ ⊗ B(K) for any Hilbert space K, we say N is completely singular in M. We prove that if N is a singular abelian von Neumann subalgebra or if N is a singular su ..."
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Cited by 3 (0 self)
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Let M be a von Neumann algebra acting on a Hilbert space H, and N be a singular von Neumann subalgebra of M. If N ¯ ⊗ B(K) is singular in M ¯ ⊗ B(K) for any Hilbert space K, we say N is completely singular in M. We prove that if N is a singular abelian von Neumann subalgebra or if N is a singular subfactor of a type II1 factor M, then N is completely singular in M. For any type II1 factor M, we construct a singular von Neumann subalgebra N of M (N ̸ = M) such that N ¯ ⊗ B(l 2 (N)) is regular (hence not singular) in M ¯ ⊗ B(l 2 (N)). If H is separable, then N is completely singular in M if and only if for any θ ∈ Aut(N ′ ) such that θ(X) = X for all X ∈ M ′ , then θ(Y) = Y for all Y ∈ N ′. As an application of this characterization of completely singularity, we prove that if M is separable (with separable predual) and N is completely singular in M, then N ¯ ⊗ ̷L is completely singular in M ¯ ⊗ ̷L for any separable von Neumann algebra ̷L.
Strong Singularity for Subfactors
, 2008
"... We examine the notion of αstrong singularity for subfactors of a II1 factor, which is a metric quantity that relates the distance between a unitary in the factor and a subalgebra with the distance between that subalgebra and its unitary conjugate. Through planar algebra techniques, we demonstrate t ..."
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Cited by 2 (0 self)
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We examine the notion of αstrong singularity for subfactors of a II1 factor, which is a metric quantity that relates the distance between a unitary in the factor and a subalgebra with the distance between that subalgebra and its unitary conjugate. Through planar algebra techniques, we demonstrate the existence of a finite index singular subfactor of the hyperfinite II1 factor that cannot be strongly singular with α = 1, in contrast to the case for masas. Using work of Popa, Sinclair, and Smith, we show that there exists an absolute constant 0 < c < 1 such that all singular subfactors are cstrongly singular. Under the hypothesis of 2transitivity, we prove that finite index subfactors are αstrongly singular with a constant that tends to 1 as the Jones Index tends to infinity and infinite index subfactors are 1strongly singular. Finally, we give a proof that proper finite index singular subfactors do not have the weak asymptotic homomorphism property relative to the containing factor. 1
MASAS AND BIMODULE DECOMPOSITIONS OF II1 FACTORS
, 812
"... Abstract. The measuremultiplicityinvariant for masas in II1 factors was introduced in [10] to distinguish masas that have the same Pukánszky invariant. In this paper we study the measure class in the measuremultiplicityinvariant. This is equivalent to studying the standard Hilbert space as an as ..."
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Cited by 2 (1 self)
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Abstract. The measuremultiplicityinvariant for masas in II1 factors was introduced in [10] to distinguish masas that have the same Pukánszky invariant. In this paper we study the measure class in the measuremultiplicityinvariant. This is equivalent to studying the standard Hilbert space as an associated bimodule. We characterize the type of any masa depending on the leftrightmeasure using Baire category methods (selection principle of Jankov and von Neumann). We present a second proof of Chifan’s result [2] and a measure theoretic proof of the equivalence of weak asymptotic homomorphism property (WAHP) and singularity that appeared in [35]. 1.
Semiregular masas of transfinite length
 Internat. J. Math
"... In 1965 Tauer produced a countably infinite family of semiregular masas in the hyperfinite II1 factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each n ∈ N, finding masas for which th ..."
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Cited by 2 (2 self)
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In 1965 Tauer produced a countably infinite family of semiregular masas in the hyperfinite II1 factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each n ∈ N, finding masas for which this procedure terminates at the nth stage. Such masas are said to have length n. In this paper we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semiregular masas in the hyperfinite II1 factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalising tower. 1
Singular and strongly mixing MASA’s in finite von
, 2008
"... Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ0,Γ) satisfies some combinatorial condition, then the abelian subalgebra A = L(Γ0) is a singular MASA in M = L(Γ) and the action of Γ0 by inner automorphisms on M satisfies the following strong mi ..."
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Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ0,Γ) satisfies some combinatorial condition, then the abelian subalgebra A = L(Γ0) is a singular MASA in M = L(Γ) and the action of Γ0 by inner automorphisms on M satisfies the following strong mixing property: for all x,y ∈ M, one has lim ‖EA(λ(γ)xλ(γ γ→∞,γ∈Γ0 −1)y) − EA(x)EA(y)‖2 = 0. Moreover, we prove that the latter property is a conjugacy invariant of the pair A ⊂ M: if θ is any automorphism of M, if the pair A ⊂ M has the strong mixing property, then so does the pair θ(A) ⊂ M. We also exhibit examples of singular MASA’s that do not satisfy the above strong mixing condition.