Relative annihilators and relative commutants in non-selfadjoint operator algebras (Q2880402)

In the paper under review, the authors introduce the concepts of the relative annihilator and relative commutant for a non-selfadjoint operator algebra and present a variety of results that can be viewed as analogous to the well-known Double Commutant Theorem of von Neumann.NEWLINENEWLINEGiven a (separable, complex) Hilbert space \({\mathcal H}\), a subspace lattice \({\mathcal L}\) is a collection of orthogonal projections onto subspaces of \({\mathcal H}\) such that \({\mathcal L}\) contains \(0\) and the identity projection \(I\) and is closed in the strong operator topology. For a given subspace lattice \({\mathcal L}\), we denote by \(\text{Alg}{\mathcal L}\) the algebra of all bounded linear operators \(A\) on \({\mathcal H}\) for which \(PAP=AP\) for all \(P\in{\mathcal L}\). Similarly, for an algebra \({\mathcal A}\) contained in \({\mathcal B}({\mathcal H})\) (the algebra of all bounded linear operators on \({\mathcal H}\)), \(\text{Lat}{\mathcal A}\) denotes the subspace lattice containing all projections \(P\) for which \(PAP=AP\) for all \(A\in{\mathcal A}\). Of special interest in this paper are algebras \({\mathcal A}\) for which the subspace lattice \(\text{Lat}{\mathcal A}\) is completely distributive and commutative. These algebras are called CDCSL algebras. The nest algebras form a particularly well-studied class of CDCSLs.NEWLINENEWLINENEWLINEFor a subset \({\mathcal S}\) of an algebra \({\mathcal A}\subseteq {\mathcal B}({\mathcal H})\) of operators on \({\mathcal H}\), the annihilator of \({\mathcal S}\) is NEWLINE\[NEWLINE{\mathcal S}^\perp =\{T\in {\mathcal B}({\mathcal H}):TS=ST=0\quad \mathrm{for all } S\in {\mathcal S}\},NEWLINE\]NEWLINE while the relative annihilator of \({\mathcal S}\) in \({\mathcal A}\) is defined by NEWLINE\[NEWLINE {\mathcal S}^\circ =\{A\in {\mathcal A}:AS=SA=0\quad\mathrm {for all } S\in {\mathcal S}\} = {\mathcal S}^\perp\cap {\mathcal A}.NEWLINE\]NEWLINE Similarly, the relative commutant of \({\mathcal S}\) in \({\mathcal A}\) is defined by NEWLINE\[NEWLINE{\mathcal S}^\dagger = \{A\in {\mathcal A}: AS=SA\quad\mathrm {for all } S\in {\mathcal S}\} = {\mathcal S}^\prime\cap {\mathcal A},NEWLINE\]NEWLINE where \({\mathcal S}^\prime\) is the usual commutant of \({\mathcal S}\) in \({\mathcal B}({\mathcal H})\). In general, \({\mathcal S}^\dagger\) contains the set \({\mathcal S}^\circ +{\mathcal Z}\), where \({\mathcal Z}\) is the center of the algebra \({\mathcal A}\). The main results of this paper are concerned with finding conditions under which \({\mathcal S}^\dagger ={\mathcal S}^\circ +\mathbf{C}I\), where \(\mathbf{C}I\) denotes the scalar operators.NEWLINENEWLINEFor example, suppose that \({\mathcal A}\) is a unital subalgebra of \({\mathcal B}({\mathcal H})\) that is equal to the closure, in the weak operator topology (WOT), of its rank-one elements and also has the property (herein called \textit{hereditarily essential}) that \({\mathcal J}_1\cap {\mathcal J}_2=\{0\}\) whenever \({\mathcal J}_1\neq{\mathcal J}_2\) are distinct WOT-closed ideals in \({\mathcal A}\). Then Theorem 2.5 shows, among other things, that \({\mathcal J}^\dagger = {\mathcal J}^\circ +\mathbf{C}I\) and \({\mathcal J}^{\dagger\dagger} = {\mathcal J}^{\circ\circ} +\mathbf{C}I\) for every WOT-closed ideal in \({\mathcal A}\). In particular, this applies whenever \({\mathcal A}\) is a nest algebra (Corollary 4.3).NEWLINENEWLINEIn Section 3 of the paper, the authors develop a variation of the characterizations due to \textit{J. A. Erdos} and \textit{S. C. Power} [J. Oper. Theory 7, 219--235 (1982; Zbl 0523.47027)] and to \textit{D.-G. Han} [Proc. Am. Math. Soc. 104, No.~4, 1067--1070 (1988; Zbl 0694.47031)] of the WOT-closed ideals of a CDCSL algebra. This characterization plays a central rôle in the proofs. Also, a CDCSL algebra known as the 4-cycle digraph algebra is used on several occasions as a counter-example to illustrate the limitations of some of the theorems.NEWLINENEWLINESection 5 focuses on algebras with a specific presentation. Namely, for two commuting orthogonal projections \(P\) and \(Q\), let \({\mathcal S}_{P,Q}=\{T\in{\mathcal B}({\mathcal H}):T=QT(I-P)\}\). More generally, suppose that \(\{P_\alpha,Q_\alpha:\alpha\in\Gamma\}\) is a set of commuting orthogonal projections and consider the algebra \({\mathcal S}=\vee\{{\mathcal S}_{P_\alpha,Q_\alpha}:\alpha\in\Gamma\}\). Setting \(P_0=\wedge P_\alpha\) and \(Q_0=\vee Q_\alpha\) as well as supposing that \({\mathcal S}_{P_\alpha,Q_\alpha}\neq 0\) for each \(\alpha\in\Gamma\), Proposition 5.4 shows that \({\mathcal S}^\perp = {\mathcal S}_{Q_0,P_0}\). With the additional technical requirement that the algebra \({\mathcal S}\) is \textit{connected} (basically meaning that the family \(\{{\mathcal S}_{P_\alpha,Q_\alpha}\}\) cannot be split into two separate subfamilies living on mutually orthogonal subspaces of \({\mathcal H}\)), Theorem 5.9 and Corollary 5.10 show that \({\mathcal S}^\prime={\mathcal S}^\perp+\mathbf{C}I\) and \({\mathcal S}^{\prime\prime}={\mathcal S}^{\perp\perp}+\mathbf{C}I\).NEWLINENEWLINEIn Section 6, the approach of Section 5 is applied in the case where \({\mathcal N}\) is a nest on \({\mathcal H}\) and the projections \(\{P_\alpha,Q_\alpha\}\) lie in \({\mathcal N}^{\prime\prime}\). In this setting, if \({\mathcal S}=\vee\{{\mathcal S}_{P_\alpha,Q_\alpha}\}\subseteq {\mathcal T}({\mathcal N})\) is connected and \({\mathcal S}_{Q_0,P_0}\neq 0\), Theorem 6.4 asserts that \({\mathcal S}^\dagger={\mathcal S}^\circ+\mathbf{C}I\) and \({\mathcal S}^{\dagger\dagger}=({\mathcal S}_{P_0,Q_0}\cap{\mathcal T}({\mathcal N}))+\mathbf{C}I\).