In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the k-folded tensor product of the free group factor (on N>1 generators). A. Sinclair and R. Smith introduced the class of strongly singular maximal abelian von Neumann algebras (MASA) of type II_1 factors. One of their examples was the Laplacian subalgebra of the free group factor, known to be a singularMASA. Using the techniques presented by A. Sinclair and R.Smith, we show that the Laplacian subalgebra of the k-folded tensor product of the free group factor with itself is a strongly singular MASA.
We prove that the radial masa C in the free group factor is disjoint from other maximal amenable subalgebras in the following sense: any distinct maximal amenable subalgebra cannot have diffuse intersection with C.
We study certain $q$-deformed analogues of the maximal abelian subalgebras of the group von Neumann algebras of free groups. The radial subalgebra is defined for Hecke deformed von Neumann algebras of the Coxeter group $(\mathbb{Z}/{2\mathbb{Z}})^{\star k}$ and shown to be a maximal abelian subalgebra which is singular and with Pukánszky invariant $\{\infty\}$. Further all non-equal generator masas in the $q$-deformed Gaussian von Neumann algebras are shown to be mutually non-unitarily conjugate.
We consider a paving property for a maximal abelian *-subalgebra (MASA) $A$ in a von Neumann algebra $M$, that we call so-paving, involving approximation in the so-topology, rather than in norm (as in classical Kadison-Singer paving). If $A$ is the range of a normal conditional expectation, then so-paving is equivalent to norm paving in the ultrapower inclusion $A^ω\subset M^ω$. We conjecture that any MASA in any von Neumann algebra satisfies so-paving. We use [MSS13] to check this for all MASAs in $\mathcal B(\ell^2 \mathbb N)$, all Cartan subalgebras in amenable von Neumann algebras and in g…
This paper studies weakly mixing (singular) and mixing masas in type $\rm{II}_{1}$ factors from a bimodule point of view. Several necessary and sufficient conditions to characterize the normalizing algebra of a masa are presented. We also study the structure of mixing inclusions, with special attention paid to masas of product class. A recent result of Jolissaint and Stalder concerning mixing masas arising out of inclusions of groups is revisited. One consequence of our structural results rules out the existence of certain Koopman-realizable measures, arising from semidirect products, which ar…
In this paper, we give examples of maximal amenable subalgebras of the free group factor of two generators. More precisely, we consider two copies of the hyperfinite factor $R_i$ of type $\mathrm{II}_1$. From each $R_i$, we take a Haar unitary $u_i$ which generates a Cartan subalgebra of it. We show that the von Neumann subalgebra generated by the self-adjoint operator $u_1+u_1^{-1}+u_2+u_2^{-1}$ is maximal amenable in the free product. This provides infinitely many non-unitary conjugate maximal amenable MASAs.
This paper illustrates the notion of a Cartan subalgebra in a C*-algebra through a number of examples and counterexamples. Some of these examples have a geometrical flavour and are related to orbifolds and non-Hausdorff manifolds.
We apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal D}_n$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal O}_n$ that are not inner conjugate to the diagonal ${\mathcal D}_n$.
In this paper we prove that for a type II_1 factor N with a Cartan maximal abelian subalgebra (masa), the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.
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.
In this paper, we exhibit strongly singular maximal abelian subalgebras living inside certain k-folded tensors of von Neumann group factors. The two classes of groups under consideration are the free groups of rank greater than 2 and the free product of m>2 groups, all finite and having the same order p>2 (m>p-1). We prove actually more, namely that the unique trace preserving conditional expectations from these group factors onto their Laplacian subalgebras are asymptotic homomorphisms, which in turn implies that the subalgebras are strongly singular masas.
We prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing, and we compute the associated bimodule. The proof relies on new properties of the Jones-Wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations.
For a Banach D-bimodule M over an abelian unital C*-algebra D, we define E(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual of M. Equip E(M) with the weak-* topology. We develop general properties of E(M). It is properly viewed as a coordinate system for M when M is a subset of C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E(C) coincides with Kumjian's twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal which contains D. For triangular subalg…
We investigate maximal abelian subalgebras (masas) in separably acting type $II_1$ factors. We use the notion of distance between masas which we introduced in an earlier paper in this archive, OA/0107075. The main result of the paper is to show that for any unitary u and a masa A, the distance between A and uAu* is comparable to the distance in 2-norm from u to the unitary normalizer N(A) of A. In qualitative terms, this says that a unitary almost normalizes A if and only if it is close to a normalizing unitary. Inequalities in the paper make this statement quantitatively precise. As a consequ…
Let $A$ be a maximal abelian subalgebra (MASA) in a \II1 factor $M$. Sorin Popa introduced an analytic condition that can be used to identify the normalizing algebra of $A$ in $M$ and which we call \emph{the relative weak asymptotic homomorphism property}. In this paper we show this property is always satisfied by the normalizing algebra of $A$ in $M$ and as a consequence we obtain that $\bar{\bigotimes}_{i\in I}(\mathcal{N}_{M_{i}}(A_{i})^{\prime\prime})= (\mathcal{N}_{\bar{\bigotimes}_{i\in I}M_{i}}(\bar{\otimes}%_{i\in I} A_{i}))^{\prime\prime}$ .
We consider an iterative procedure for constructing maximal abelian $^*$-subalgebras (MASAs) satisfying prescribed properties in II$_1$ factors. This method pairs well with the intertwining by bimodules technique and with properties of the MASA and of the ambient factor that can be described locally. We obtain such a local characterization for II$_1$ factors $M$ that have an {\it s-MASA}, $A\subset M$ (i.e., for which $A \vee JAJ$ is maximal abelian in $\Cal B(L^2M)$), and use this strategy to prove that any factor in this class has uncountably many non-intertwinable singular (respectively sem…
We show that any type ${\rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $\ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type ${\rm III_1}$ factors.
Two classical theorems in matrix theory, due to Schur and Horn, relate the eigenvalues of a self-adjoint matrix to the diagonal entries. These have recently been given a formulation in the setting of operator algebras as the Schur-Horn problem, where matrix algebras and diagonals are replaced respectively by finite factors and maximal abelian self-adjoint subalgebras (masas). There is a special case of the problem, called the carpenter problem, which can be stated as follows: for a masa A in a finite factor M with conditional expectation E_A, can each x in A with 0 <= x <= 1 be expressed as E_…
Let $Γ$ be a countable group and let $Γ_0$ be an infinite abelian subgroup of $Γ$. We prove that if the pair $(Γ,Γ_0)$ satisfies some combinatorial condition called (SS), then the abelian subalgebra $A=L(Γ_0)$ 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 satisf…
The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\ \bar{\otimes}\ B_2$ is maximal injective…
In this paper we exploit the fact that a Cuntz C$^*$-algebra is a groupoid C$^*$-algebra to facilitate the study of non-self-adjoint subalgebras of $O_n$. The Cuntz groupoid is not principal and the spectral theorem for bimodules does not apply in full generality. We characterize the bimodules (over a natural masa) which are determined by their spectra in the Cuntz groupoid; these are exactly the ones which are invariant under the guage automorphisms and exactly the ones which are generated by the Cuntz partial isometries which they contain. We investigate analytic subalgebras of $O_n, n$ fini…
We consider two von Neumann subalgebras $\cl B_0$ and $\cl B$ of a type ${\rm{II}}_1$ factor $\cl N$. For a map $ϕ$ on $\cl N$, we define \[\|ϕ\|_{\infty,2}=\sup\{\|ϕ(x)\|_2\colon \|x\| \leq 1\},\] and we measure the distance between $\cl B_0$ and $\cl B$ by the quantity $\|{\bb E}_{\cl B_0}-{\bb E}_{\cl B}\|_{\infty,2}$. Under the hypothesis that the relative commutant in $\cl N$ of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the $\|\cdot \|_2$--norm. This hypothesis is satisfie…
We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups; in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we constr…
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…
This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the first free entropy proof of Popa's famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer's results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if $\mathcal{M} = \mathcal{P} * \mathcal{Q}$, then $\mathcal{P}$ absorbs any subalgebra of $\mathcal{M}$ that intersects it diffusely …
Automorphisms of the canonical core UHF-subalgebra F_n of the Cuntz algebra O_n do not necessarily extend to automorphisms of O_n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras M_n. In that case, necessary and sufficient conditions for the extension property are presented. It is also addressed the problem of extending to O_n the automorphisms of the diagonal D_n, which is a regular MASA with Cantor spectrum. In particular, it is shown the existence of product-type automorphisms of D_n that are not extensible to (poss…
We examine the notion of $α$-strong singularity for subfactors of a \IIi 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 \IIi 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 $c…
