The set of normalizers between von Neumann (or, more generally, reflexive) algebras A and B, (that is, the set of all operators x such that xAx* is a subset of B and x*Bx is a subset of A) possesses `local linear structure': it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces U for which UU*U is a subset of U. Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form U={x:xp=h(p)x, for all p in P}, where P is a set of projections and h a certain map defined on P. A normalizing space consists of normalizers between appropriate von Neumann algebras A and B. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian selfadjoint algebras consist of operators `supported' on sets of the form [f=g] where f and g are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.
We investigate subalgebras $A$ of the Cuntz algebra ${\mathcal O}_n$ that arise as finite direct sums of corners of the UHF-subalgebra ${\mathcal F}_n$. For such an $A$, we completely determine its normalizer group inside ${\mathcal O}_n$.
As a generalization of the free semigroup algebras considered by Davidson and Pitts, and others, the second author and D.W. Kribs initiated a study of reflexive algebras associated with directed graphs. A free semigroupoid algebra L_G is generated by a family of partial isometries, and initial projections, acting on a generalized Fock space spawned from the directed graph G. We show that if the graph is finite then the algebra L_G is hyper-reflexive.
In this paper we investigate the normalizer $\mathcal{N}_{\mathcal{O}_{n}}(A)$ of a $C^{*}$-subalgebra $A\subset \mathcal{F}_{n}$ where $\mathcal{F}_{n}$ is the canonical UHF-subalgebra of type $n^{\infty}$ in the Cuntz algebra $\mathcal{O}_{n}$. Under the assumption that the relative commutant $A'\cap \mathcal{F}_{n}$ is finite-dimensional, we show several facts for normalizers of $A$. In particular it is shown that the automorphism group $\{{\rm Ad}u|_{A}\ \ |\ u\in \mathcal{N}_{\mathcal{F}_{n}}(A)\}$ has a finite index in $\{{\rm Ad}U|_{A}\ \ |\ U\in \mathcal{N}_{\mathcal{O}_{n}}(A)\}$.
Let $H^{\infty}(E)$ be a non commutative Hardy algebra, associated with a $W^*$-correspondence $E$. These algebras were introduced in 2004, ~\cite{MuS3}, by P. Muhly and B. Solel, and generalize the classical Hardy algebra of the unit disc $H^{\infty}(\mathbb{D})$. As a special case one obtains also the algebra $\mathcal{F}^{\infty}$ of Popescu, which is $H^{\infty}(\mathbb{C}^n)$ in our setting. In this paper we view the algebra $H^\infty(E)$ as acting on a Hilbert space via an induced representation $ρ(H^{\infty}(E))$, and we study the reflexivity of $ρ(H^{\infty}(E))$. This question was stu…
Given a w*-closed unital algebra $A$ acting on $H_0$ and a contractive w*-continuous endomorphism $β$ of $A$, there is a w*-closed (non-selfadjoint) unital algebra $\mathbb{Z}_+\bar{\times}_βA$ acting on $H_0\otimes\ell^2({\mathbb{Z}_+})$, called the w*-semicrossed product of $A$ with $β$. We prove that the w*-semicrossed product is a reflexive operator algebra provided $A$ is reflexive and $β$ is unitarily implemented, and that it has the bicommutant property if and only if so does $A$. Also, we show that the w*-semicrossed product generated by a commutative C*-algebra and a *-endomorphism is…
We consider an inclusion $B\subseteq M$ of finite von Neumann algebras satisfying $B'\cap M\subseteq B$. A partial isometry $v\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\subseteq B$. Given two such inclusions $B_i\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \vnotimes B_2$ in $M_1\vnotimes M_2$, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail…
Let X be a compact Hausdorff space and let H be a separable Hilbert space. We prove that the group of all order automorphisms of the $C^*$-algebra $C(X)\otimes B(H)$ is algebraically reflexive.
A free semigroup algebra S is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for S if the set of images of x under non-commuting words in the generators of S is orthonormal. We establish the following dichotomy: either a free semigroup algebra has a wandering vector, or it is a von Neumann algebra. Consequences include that every free semigroup algebra is reflexive, and that certain free semigroup algebras are hyper-reflexive with a very small hyper-reflexivity constant.
We discuss different mixing properties for triples of finite von Neumann algebras $B\subset N\subset M$, and we introduce families of triples of groups $H<K<G$ whose associated von Neumann algebras $L(H)\subset L(K)\subset L(G)$ satisfy $\mathcal{N}_{L(G)}(L(H))"=L(K)$. It turns out that the latter equality is implied by two conditions: the equality $\mathcal{N}_G(H)=K$ and the above mentioned mixing properties. Our families of examples also allow us to exhibit examples of pairs $H<G$ such that $L(\mathcal{N}_G(H))\not=\mathcal{N}_{L(G)}(L(H))"$.
We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products over factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bico…
A C*-algebra $A$ is C*-reflexive if any countably generated Hilbert C*-module $M$ over $A$ is C*-reflexive, i.e. the second dual module $M''$ coincides with $M$. We show that a commutative C*-algebra $A$ is C*-reflexive if and only if for any sequence $I_k$ of disjoint non-zero C*-subalgebras, the canonical inclusion $\oplus_k I_k\subset A$ doesn't extend to an inclusion of $\prod_k I_k$.
The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.
The operator space analogue of the {\em strong form} of the principle of local reflexivity is shown to hold for any von Neumann algebra predual, and thus for any $C^{*}$-algebraic dual. This is in striking contrast to the situation for $C^{*}$-algebras, since, for example, $K(H)$ does not have that property. The proof uses the Kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces.
A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra $\mathcal{D}_2$ in the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$ is proved not to be regular. Finally, countably many novel permutative endomorphisms of $\mathcal{Q}_2$ are exhibited with prescribed images of the generator $U$.
We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation which gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of $H^{\infty}(\bb{T})\otimes\cl B(\cl H)$.
A triple of finite von Neumann algebras $B\subseteq N\subseteq M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators $\{u_λ\}_{λ\in Λ}$ in $B$ such that $$\lim_λ|\mathbb{E}}_B(xu_λy)-{\mathbb{E}}_B({\mathbb{E}}_N(x)u_λ{\mathbb{E}}_N(y))\|_2=0$$ for all $x,y\in M$. We prove that a triple of finite von Neumann algebras $B\subseteq N\subseteq M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subseteq \sum_{i=1}^n x_iB$ for a finite number of elements $x_1,...,x_n$ i…
We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated to generalized box spaces.
Recently, Adrian Ioana proved that all crossed products by free ergodic probability measure preserving actions of a nontrivial free product group Γ_1 * Γ_2 have a unique Cartan subalgebra up to unitary conjugacy. Ioana deduced this result from a more general dichotomy theorem on the normalizer N_M(A)'' of an amenable subalgebra A of an amalgamated free product von Neumann algebra M = M_1 *_B M_2. We improve this dichotomy theorem by removing the spectral gap assumptions and obtain in particular a simpler proof for the uniqueness of the Cartan subalgebra in crossed products by Γ_1 * Γ_2.
We introduce and study the notion of null-orbit reflexivity, which is a slight perturbation of the notion of orbit-reflexivity. Positive results for orbit reflexivity and the recent notion of $\mathbb{C}$-orbit reflexivity both extend to null-orbit reflexivity. Of the two known examples of operators that are not orbit-reflexive, one is null-orbit reflexive and the other is not. The class of null-orbit reflexive operators includes the classes of hyponormal, algebraic, compact, strictly block-upper (lower) triangular operators, and operators whose spectral radius is not 1. We also prove that eve…
We study quasidiagonality and local reflexivity for $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. We introduce and study a notion of amenability for vector valued traces.
Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a structure theory for the weak operator topology closed algebras generated by these representations, which we call free semigroupoid algebras. We characterize semisimplicity in terms of the graph and show explicitly in the case of finite graphs how the Jacobson radical is determined. We provide a diverse collection of examples including; algebras with free behaviou…
S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer N of SU(1,1) in SL(2,C) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing N_q, a new example of a unimodular locally compact quantum group (depending on a parameter q) that is a deformation of N. After defining the underlying von Neumann algebra of N_q we use a certain class of q-hypergeometric functions an…
The quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is an important and non-trivial example of a non-compact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of the paper is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra U_q(su(1,1)). It turns out that U_q(su(1,1)) does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommut…
Many interesting examples of operator algebras, both self-adjoint and non-self-adjoint, can be constructed from directed graphs. In this survey, we overview the construction of $C^*$-algebras from directed graphs and from two generalizations of graphs: higher rank graphs and categories of paths. We also look at free semigroupoid algebras generated from graphs and higher rank graphs, with an emphasis on the left regular free semigroupoid algebra. We give examples of specific graphs and the algebras they generate, and we discuss properties such as semisimplicity and reflexivity. Finally, we prop…
We provide a direct and elementary proof of the equivalence between the weak asymptotic homomorphism property for the pair of group von Neumann algebras $L(H)\subset L(G)$ and the embedding into $H$ of the one sided quasi-normalizer of the pair $H<G$, as it is stated by J. Fang, M. Gao and R. R. Smith in a recent article, and we extend the result to triples of groups $H<K<G$.
In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant.
Noncommutative multivariable versions of weighted shifts arise naturally as `weighted' left creation operators acting on Fock space. We investigate the unital weak operator topology closed algebras they generate. The unweighted case yields noncommutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the noncommutative setting. We prove these algebras are reflexive when the eigenvalues for the adjoint algebra include an open set in complex $n$-space, and provide a new elementary proof of …
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 begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a 'higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs …
Very recently, Špakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity. In this paper, we improve their method to deal with the $L^p$-version of (uniform) Roe algebras for any $p\in [1,\infty)$. Due to the lack of reflexivity on $L^1$-spaces, some extra work is required for the case of $p=1$.
We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $ϕ:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $ϕ$ is an automorphism of the algebra B(H).
We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.
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 $f\in\ell_2(Γ)$ which act by left convolution on $\ell_2(Γ)$. Let $T^r$ be a totally geodesic flat torus of dimension $r$ in $M$ and let $Γ_0\cong\bb Z^r$ be the image of the fundamental group of $T^r$ in $Γ$. Then $\vn(Γ_0)$ is a maximal abelian $\star$-subalgebra of $\vn(Γ)$ and its unitary normalizer is as small as possible. If $M$ has constant negative curvature then the Pukánszky invariant of $\vn(Γ_0)$ is $\infty$.
For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that …
