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 $M_1\ \bar{\otimes}\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors.
We present families of pairs of finite von Neumann algebras $A\subset M$ where $A$ is a maximal injective masa in the type $\mathrm{II}_1$ factor $M$ with separable predual. Our results make use of the strong mixing and the asymptotic orthogonality properties of $A$ in $M$ and are borrowed from ideas of S. Popa who proved that if $G$ is a non abelian free group and if $a$ is one of its generators, then the von Neumann algebra generated by $a$ is maximal injective in the factor $L(G)$. Our results apply to pairs $H<G$ where $H$ is an infinite abelian subgroup of a suitable amalgamated product g…
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.
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.
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 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…
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.
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 …
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…
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 $A$ 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 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.
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…
A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes_{\inj}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^*$-algebras; this is a sort of a `dual' result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum-Connes conjecture. It turns out that $\r…
We generalize the notion of `diagonal' from the class of CSL algebras to masa bimodules. We prove that a reflexive masa bimodule decomposes as a sum of two bimodules, the diagonal and a module generalizing the w*-closure of the Jacobson radical of a CSL algebra. The latter module turns out to be reflexive, a result which is new even for CSL algebras. We show that the projection onto the direct summand contained in the diagonal is contractive and preserves compactness and reduces rank of operators. Stronger results are obtained when the module is the reflexive hull of its rank-one subspace. Key…
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.
In this article, we proved the following results. Let $L(F(n_i))$ be the free group factor on $n_i$ generators and $λ(g_{i})$ be one of standard generators of $L(F(n_i))$ for $1\le i\le N$. Let $\A_i$ be the abelian von Neumann subalgebra of $L(F(n_i))$ generated by $λ(g_{i})$. Then the abelian von Neumann subalgebra $\otimes_{i=1}^N\A_i$ is a maximal injective von Neumann subalgebra of $\otimes_{i=1}^N L(F(n_i))$. When $N$ is equal to infinity, we obtained McDuff factors that contain maximal injective abelian von Neumann subalgebras.
Let M_i be a von Neumann algebra, and B_i be a maximal injective von Neumann subalgebra of M_i, i=1,2. If M_1 has separable predual and the center of B_1 is atomic, e.g., B_1 is a factor, then B_1\tensor B_2 is a maximal injective von Neuamnn subalgebra of M_1\tensor M_2. This partly answers a question of Popa
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…
A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.
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…
We study Borel homomorphisms $θ: G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$θ_*(μ)(α)=μ(θ^{-1}(α))\quad \text{for } \quad α\subseteq H $$ is absolutely continuous with respect to $ν,$ where $μ$ (resp. $ν$) is a Haar measure for $G,$ (resp. $H$). We define a natural mapping $\mathcal G$ from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in $B(L^2(H))$ into the class of masa bimodules in $B(L^2(G))$ and we use it to prove that if $k\subseteq G\times G$ is a set of operator synthesis, then $(θ\times θ)^{-…
For a given discrete group $G$, we apply results of Kirchberg on exact and injective tensor products of $C^*$-algebras to give an explicit description of the minimal exact correspondence crossed-product functor and the maximal injective crossed-product functor for $G$ in the sense of Buss, Echterhoff and Willett. In particular, we show that the former functor dominates the latter.
In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.
In this paper, for a C*-Algebra A with M = M(A) an AW*-algebra, or equivalently, for an essential, norm-closed, two-sided ideal A of an AW*-algebra M, we investigate the strict approximability of the elements of M from commutative C*- subalgebras of A. In the relevant case of the norm-closed linear span A of all finite projections in a semi-finite AW*-algebra M we shall give a complete description of the strict closure in M of any maximal abelian self-adjoint subalgebra (masa) of A. We shall see that the situation is completely different for discrete respectively continuous M : in the discrete…
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 undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphis…
