Maximal Abelian Self-adjoint Algebra Research Articles

Schur Multipliers of Cartan Pairs

AbstractWe define the Schur multipliers of a separable von Neumann algebrawith Cartan maximal abelian self-adjoint algebra, generalizing the classical Schur multipliers of(ℓ2). We characterize these as the normal-bimodule maps on. Ifcontains a direct summand isomorphic to the hyperfinite II1factor, then we show that the Schur multipliers arising from the extended Haagerup tensor product⊗ehare strictly contained in the algebra of all Schur multipliers.

Bilattices and Morita Equivalence of MASA Bimodules

AbstractWe define an equivalence relation between bimodules over maximal abelian self-adjoint algebras (MASA bimodules), which we call spatial Morita equivalence. We prove that two reflexive MASA bimodules are spatially Morita equivalent if and only if their (essential) bilattices are isomorphic. We also prove that if are bilattices that correspond to reflexive MASA bimodules , and is an onto bilattice homomorphism, then(i) if is synthetic, then is synthetic;(ii) if contains a non-zero compact (or a finite or a rank 1) operator, then also contains a non-zero compact (or a finite or a rank 1) operator.

Normalisers of operator algebras and tensor product formulas

We establish a tensor product formula for bimodules over maximal abelian selfadjoint algebras and their supports. We use this formula to show that if A is the tensor product of finitely many continuous nest algebras, B is a CSL algebra and A and B have the same normaliser semi-group then either A = B or A∗ = B. We show that the result does not hold without the assumption that the nests be continuous, answering in the negative a question raised in [28].

On almost-invariant subspaces and approximate commutation

A closed subspace Y of a Banach space X is almost-invariant for a collection S of bounded linear operators on X if for each T∈S there exists a finite-dimensional subspace FT of X such that TY⊆Y+FT. In this paper, we study the existence of almost-invariant subspaces for algebras of operators. We show, in particular, that if a closed algebra of operators on a Hilbert space has a non-trivial almost-invariant subspace then it has a non-trivial invariant subspace. We also examine the structure of operators which admit a maximal commuting family of almost-invariant subspaces. In particular, we prove that if T is an operator on a separable Hilbert space and if TP−PT has finite rank for all projections P in a given maximal abelian self-adjoint algebra M then T=M+F where M∈M and F is of finite rank.

On the continuity of the support of bimodules over maximal abelian self-adjoint algebras

We relate the convergence of a net of maximal Abelian selfadjoint algebras (masas) to that of the net of their corresponding supports. This is achieved by using a family of capacities on the collection of subsets of X × Y (where the masas are realized as collections of operators of multiplication by essentially bounded functions on the measure spaces X and Y), which extends a capacity studied previously by Haydon and Shulman.

NORMALIZERS OF OPERATOR ALGEBRAS AND REFLEXIVITY

The set of normalizers between von Neumann (or, more generally, reflexive) algebras $\mathcal{A}$ and $\mathcal{B}$ (that is, the set of all operators $T$ such that $T \mathcal{A} T^{\ast} \subseteq \mathcal{B}$ and $T^{\ast} \mathcal{B} T \subseteq \mathcal{A}$) 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 $\mathcal{U}$ of operators satisfying $\mathcal{UU}^{\ast} \mathcal{U} \subseteq \mathcal{U}$ (also known as ternary rings of operators). Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form $\mathcal{U} = \{T : TL = \phi (L) T$ for all $L \in \mathcal{L}\}$ where $\mathcal{L}$ is a set of projections and $\phi$ a certain map defined on $\mathcal{L}$. A normalizing space consists of normalizers between appropriate von Neumann algebras $\mathcal{A}$ and $\mathcal{B}$. 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 self-adjoint 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.2000 Mathematical Subject Classification: 47L05 (primary), 47L35, 46L10 (secondary).

The Uniqueness of AF Diagonals in Regular Limit Algebras

Necessary and sufficient conditions are obtained for the uniqueness of standard regular AF maximal abelian self-adjoint algebras in regular limit algebras up to approximate inner unitary equivalence.

Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras

Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology. A new method of representing masa bimodules is introduced. This uses a novel concept of anω-topology. With the appropriate notion ofω-support, a correspondence is established between reflexive masa bimodules and theirω-supports. It is shown that, if a C2-closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C2-norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to Cpfor allp>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in Cpfor 1⩽p<∞. As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.

On the Closure of Triangular Algebras

algebras of operators on Hilbert space introduced by Kadison and Singer in [8]. If sA is a maximal abelian self-adjoint algebra (m.a.s.a.) in B(NJ) then an algebra of operators, 9T, is said to be triangular with diagonal, S, if S-T n o* = s4. This definition transfers naturally to a subalgebra, T, of a von Neumann algebra, M, for which T n T* is a m.a.s.a. in M. Recently, Muhly, Saito and Solel [9] have made an extensive study of a-weakly closed triangular subalgebras of von Neumann algebras when the diagonal m.a.s.a. is a Cartan subalgebra of M. In particular, they have obtained a structure theorem for cr-weakly closed maximal triangular algebras which generalizes the most regular behaviour of triangular algebras in B(NW). However, the most general results on triangular algebras (by which we mean results for which as little as possible is assumed about the algebra beyond its triangularity) have proved elusive. One natural question of this type, which was raised by Erdos in [5], is; Question 1. Is the norm closure of a triangular algebra necessarily triangular? It follows from a straightforward Zorn's Lemma argument that every triangular algebra in B(C) (resp. a von Neumann algebra, M) is contained in a maximal triangular algebra. Thus, Question 1 is equivalent to,

On Derivations of Certain Algebras Related to Irreducible Triangular Algebras

This paper deals with derivations on algebras that are generated by a maximal abelian selfadjoint algebra of operators $\mathcal {A}$ on a Hilbert space and a group of unitary operators acting on it. A necessary and sufficient condition for such a derivation to be implemented by an operator affiliated with $\mathcal {A}$ is given. The results are related to the study of derivations on a certain class of irreducible triangular algebras.

On derivations of certain algebras related to irreducible triangular algebras

This paper deals with derivations on algebras that are generated by a maximal abelian selfadjoint algebra of operators A \mathcal {A} on a Hilbert space and a group of unitary operators acting on it. A necessary and sufficient condition for such a derivation to be implemented by an operator affiliated with A \mathcal {A} is given. The results are related to the study of derivations on a certain class of irreducible triangular algebras.

The radical of a reflexive operator algebra

The radical of a reflexive operator algebra % whose lattice of invariant subspaces 2 is commutative is related to the space of lattice homomorphisms of 2 onto {0,1}. To each such homomorphism φ is associated a closed, two-sided ideal Wψ contained in H. The intersection of the ^ is contained in the radical; it is conjectured that equality always holds. The conjecture is proven for a variety of special cases: countable direct sums of nest algebras; finite direct sums of algebras which satisfy the conjecture; algebras whose lattice of invariant subspaces is finite; algebras whose lattice of invariant subspaces is isomorphic to the lattice of nonincreasing sequences with values in N U {°°}. 1* Introduction* This paper studies the radical of a certain class of non-self-ad joint operator algebras. Given an algebra 21 and a lattice S of orthogonal projections acting on a separable Hubert space φ, we use the standard notations, Sat 21 and SttgS, to denote, respectively, the lattice of all projections invariant under Sί and the algebra of all (bounded) operators which leave invariant each projection of 2. 2t and 2 are said to be reflexive if 2ί = Stlg Sat 2C and S = Sat 2CIg S, respectively. The algebras which we study are reflexive algebras which contain a maximal abelian self-adjoint algebra (m.a.s.a.). A commutative subspace lattice is a lattice of pairwise commuting, orthogonal projections on $ which contains 0 and 1 and which is closed in the strong operator topology. It follows automatically that a commutative subspace lattice is a complete lattice. If 2C is an operator algebra containing a m.a.s.a., then Sat SI is a commutative subspace lattice. Every commutative subspace lattice, S, is reflexive ([1], p. 468), and SίlgS is a reflexive algebra which contains a m.a.s.a. Henceforth, all lattices of projections in this paper will be commutative subspace lattices and all algebras will be reflexive algebras which contain a m.a.s.a. An incisive study of these lattices and algebras by Arveson is found in [1]. At least in certain special cases, the radical of a reflexive algebra, 2ί, containing a m.a.s.a. can be described in terms of the set of lattice homomorphisms from S = Sαt2t onto {0,1}. To each such homomorphism φ we shall associate a closed two-sided ideal %φ in St. The radical, % of Sf is equal to the intersection of these ideals. It appears reasonable to conjecture that this equality holds for all

On Operator Algebras and Invariant Subspaces

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.