Jon P. Bannon, Mohan Ravichandran
In this article we introduce an isomorphism invariant for type II_1 factors using the Connes-Folner condition. We compute bounds of this number for free group factors.
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Let $G$ be a connected semisimple real algebraic group. Assume that $G(\bb R)$ has no compact factors and let $Γ$ be a torsion-free uniform lattice subgroup of $G(\bb R)$. Then $Γ$ contains a malnormal abelian subgroup $A$. This implies that the $\tto$ factor $\vn(Γ)$ contains a masa $\fk A$ with Pukánszky invariant $\{\infty\}$.
In 1960 Pukánszky introduced an invariant associating to every masa in a separable $\mathrm{II}_1$ factor a non-empty subset of $\mathbb N\cup\{\infty\}$. This invariant examines the multiplicity structure of the von Neumann algebra generated by the left-right action of the masa. In this paper it is shown that every non-empty subset of $\mathbb N\cup\{\infty\}$ arises as the Pukánszky invariant of some masa in a separable McDuff $\mathrm{II}_1$ factor which contains a masa with Pukánszky invariant $\{1\}$. In particular the hyperfinite $\mathrm{II}_1$ factor and all separable McDuff $\mathrm{I…
We show that any amenable von Neumann subalgebra of any free Araki-Woods factor that is globally invariant under the modular automorphism group of the free quasi-free state is necessarily contained in the almost periodic free summand.
We investigate certain classes of normal completely positive (CP) maps on the hyperfinite $II_1$ factor $\mathcal A$. Using the representation theory of a suitable irrational rotation algebra, we propose some computable invariants for such CP maps.
Let A be a maximal abelian self-adjoint subalgebra (masa) in a type II_1 factor M acting via standard representation on L^2(M). The abelian von Neumann algebra A generated by A and JAJ has a type I commutant which contains the projection e_A from L^2(M) onto L^2(A). Then A'(1-e_A) decomposes into a direct sum of type I_n algebras for n in {1,2,...,\infty}, and those n's which occur in the direct sum form a set called the Pukanszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S\cup {\infty} whe…
In \cite{Ioana:vNsuperrigidity}, Ioana introduced three new invariants of type II$_1$ factors: the one-sided fundamental group, the endomorphism semigroup and the set of right-finite bimodules. In \cite{Ioana:vNsuperrigidity}, he does not provide many computations of these invariants. In particular, the question whether these invariants can be trivial is left open. We give an explicit example of a type II$_1$ factor for which all three invariants are trivial. More generally, for any countable left-cancellative semigroup $G$, we construct a type II$_1$ factor $M$ whose endomorphism semigroup is…
We investigate factoriality, Connes' type ${\rm III}$ invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type ${\rm III}$ invariants.
We introduce a new invariant \mathcal{S}(M) for type III factors M with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant τcannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary self adjoint operators. (2) A McDuff factor embeds into the ultrapower of the hyperfinite II_1 factor if and only if it contains a weakly dense operator system which is injective. (3) There exists a simple, quasidiagonal, real rank zero C*-algebra with non-hyperfinite II_1 factor representations and which is not tracially AF. This answers negatively questio…
We prove that a type II$_1$ factor $M$ can have at most one Cartan subalgebra $A$ satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class $\Cal H \Cal T$ of factors $M$ having such Cartan subalgebras $A \subset M$, the Betti numbers of the standard equivalence relation associated with $A \subset M$ ([G2]), are in fact isomorphism invariants for the factors $M$, $β^{^{HT}}_n(M), n\geq 0$. The class $\Cal H\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $β^{^{HT}}_n(M^t)= β^{^{HT}}_n…
We define the standard Borel space of free Araki-Woods factors and prove that their isomorphism relation is not classifiable by countable structures. We also prove that equality of $τ$-topologies, arising as invariants of type III factors, as well as coycle and outer conjugacy of actions of abelian groups on free product factors are not classifiable by countable structures.
Let $\M$ be a semi-finite factor and let $\J(\M)$ be the set of operators $T$ in $\M$ such that $T=ETE$ for some finite projection $E$. In this paper we obtain a representation theorem for unitarily invariant norms on $\J(\M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $\J(\M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical result \cite{vN} on unitarily invariant norms on $M_n(\cc)$. As another application, Ky Fan's dominance theorem \cite{Fan} is obtained for …
We show that for any prime p and for any II$_1$ factor N there exist two $mathbb{Z}_{p^2}$-- actions on the free product factor $*_{1} ^{p}N$ that have the same outer invariant but are not outer conjugate. Therefore, in the case of free product factors, the outer invariant is not a complete invariant for outer conjugacy.
Classification of Invariant Subalgebras in a class of factors with property (T)
本文研究了具有性质(T)的群$G_n$在因子$L(G_n)$中的不变子代数分类，针对$n=2$和$n\geq 3$分别给出了不同结论，并推导出$L(G_n)$具有唯一最大Haagerup不变子代数的结论。
We construct numerous continuous families of irreducible subfactors of the hyperfinite II$_1$ factor, which are non-isomorphic, but have all the same standard invariant. In particular, we obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II$_1$ factor, which have Jones index 6 and have all the same standard invariant with property (T). We exploit the fact that property (T) groups have uncountably many non-cocycle conjugate cocycle actions on the hyperfinite II$_1$ factor.
We calculate Connes' invariant $χ(M)$ for certain II$_1$ factors $M$ that can be obtained as inductive limits of subfactors with spectral gap, then use this to answer a question he posed in 1975, on the structure of McDuff factors $M$ with $χ(M)=1$.
We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the $1$-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II${}_1$-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II${}_1$-factor carries…
This revised version corrects some substancial errors in the proofs. The old proofs were only valid for (finite dimensional) * (finite dimensional). The scope of the results remains the same, however for type I_\infty algebras, the results of the original version were incorrect.
Fullness and Connes' $\boldsymbolτ$ invariant of type III tensor product factors
本文证明了任意两个满因子的张量积仍为满因子，计算了其Connes $τ$不变量，揭示了III型满因子的增强与最优谱隙性质，并证明了其与非I型顺从因子张量积的唯一McDuff分解。
We associate a cohomological invariant to each outer action of a group on a factor, and classify them by the invariant in the case that the group is a countable discrete amenable group and the factor is appoximately finite dimensional. The invariant defined for the group Out(M)=Aut(M)/Int(M) is called the intrinsic modular obstruction. The invariant for an outer action alpha is given as the pull back of the intrinsic modular obstruction, which is called the modular obstruction of alpha and denoted by Ob_m(alpha). This is the first part of the theory and presents general theory. In the case tha…
We will study two kinds of actions of a discrete amenable Kac algebra. The first one is an action whose modular part is normal. We will construct a new invariant which generalizes a characteristic invariant for a discrete group action, and we will present a complete classification. The second is a centrally free action. By constructing a Rohlin tower in an asymptotic centralizer, we will show that the Connes-Takesaki module is a complete invariant.
We introduce four new cocycle conjugacy invariants for E_0-semigroups on II_1 factors: a coupling index, a dimension for the gauge group, a super product system and a C*-semiflow. Using noncommutative Itô integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles. We do this for the Clifford flows and even Clifford flows on the hyperfinite II_1 factor, and for the free flows on the free group factor $L(F_\infty)$. In all cases the index is 0, which implies they have trivial gauge groups. We compute the super product systems for these fam…
Using methods of R.J.Tauer we exhibit an uncountable family of singular masas in the hyperfinite $\textrm{II}_1$ factor $\R$ all with Pukánszky invariant $\{1\}$, no pair of which are conjugate by an automorphism of $R$. This is done by introducing an invariant $Γ(A)$ for a masa $A$ in a \IIi factor $N$ as the maximal size of a projection $e\in A$ for which $A e$ contains non-trivial centralising sequences for $eN e$. The masas produced give rise to a continuous map from the interval $[0,1]$ into the singular masas in $\R$ equiped with the $d_{\infty,2}$-metric. A result is also given showing …
The measure-multiplicity-invariant for masas in $\rm{II}_{1}$ factors was introduced in \cite{MR2261688} to distinguish masas that have the same Pukánszky invariant. In this paper we study the measure class in the measure-multiplicity-invariant. This is equivalent to studying the standard Hilbert space as an associated bimodule. We characterize the type of any masa depending on the left-right-measure using Baire category methods (selection principle of Jankov and von Neumann). We present a second proof of Chifan's result on normalisers and a measure theoretic proof of the equivalence of weak a…
Relative solidity results and their applications to computations of some II$_1$ factor invariants
本文证明相对双曲群的冯·诺依曼代数具有相对固体性，并据此构造连续统个两两非虚拟同构的icc性质(T)因子，其单侧基本半群平凡。
Let Q be any II_1-factor. It is shown that any standard lattice G can be realized as the standard invariant of a free product of (several) rescalings of Q. In particular, if Q has fundamental group equal to the positive reals and if P is the free product of infinitely many copies of Q, then P has subfactors giving rise to all possible standard invariants. Similarly, given a II_1-subfactor N of M, it is shown there is a subfactor R of S having the same standard invariant as N in M but where S, respectively R, is the free product of M, respectively N, with rescalings of Q.
It is shown that to every operator T in a general von Neumann factor M of type II_1 and to every Borel set B in the complex plane, one can associate a largest, closed, T-invariant subspace, K = K_T(B), affiliated with M, such that the Brown measure of T|_K is concentrated on B. Moreover, K is T-hyperinvariant, and the Brown measure of (1-P_K)T|_(1-P_K)(H) is concentrated on C\B. In particular, if T has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T in M, the limit A=\lim_{n\…
In this paper we introduce and study strongly singular maximal abelian self-adjoint subalgebras of type $II_1$ factors. We show that certain elements of free groups and of non-elementary hyperbolic groups generate such masas, and these also give new examples of masas for which Popa's invariant delta is 1. We also explore the connection between Popa's invariant and strong singularity.
A new construction of factors of type ${\rm III_1}$
本文构造了一族新的III₁型因子，并在一定条件下借助Popa的结果给出完全分类，同时构造出具有相同τ不变量但两两不同构的不可数III₁型因子族。
$k$-Regular Factorizations and Invariant Subspaces of Completely Non-Unitary Contractions
引入压缩算子的$k$-正则分解概念，建立其与完全非酉压缩的不变子空间链之间的一一对应，并构造显式泛函模型，同时引入交换$k$-压缩的对称$k$-正则元组，证明经典反例不满足该性质。
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