Reflexive Algebras

This thesis is devoted to the investigation of some problems concerning reflexive oper-ator algebras. It consists of eight chapters.In Chapter 1, we introduce some terminology and notation, and summarize the back-ground and the main contents of this paper.In Chapter 2. we discuss the order homomorphisms and preannihilators of reflexive modules over reflexive operator algebras.Let A be a reflexive operator algebra acting on a Hilbert space H, U be any reflexive A-module, and φ be an order homomorphism determining U , i.e., φ is an order-preserving mapping from Lat.4 into itself such that We obtain the following results.(1) If K is a closed subspace of H, then K ∈ LatU if and only if there exists some E ∈ Lat A such that φ (E) K E . Here U denotes the preannihilator of U in the ideal T(H) of all trace class operators, and φ^E= V {F ∈ Lat A : φ(F) E} .(2) If A is a nest algebra, then LatA is maximal if and only if Lat A = LatA(3) If A is a reflexive operator algebra which is weakly generated by rank one operators, then the weakly closures of U and (U) are the A-modules determined by φ and φ, respectively .(4) Let A be as in (3). Then φ is the least homomorphism determining U if and only if (φ⁾ = φ(5) [13, Corollary 23.5] (its dual is [59, Proposition 3.2]) and [60, Theorem 3.1] are recaptured as corollaries of our main results.In Chapter 3. we first present the decomposition of finite rank operators and trace class operators in a weakly closed nest algebra AlgN-module. Secondly, the same question for finite rank operators in the preanm'hilator of an atomic Boolean lattice algebra is considered.(6) Let N be a nest on a Hilbert space H. and U be any weakly closed AlgN-module. If T ∈ U is of rank n , then there are n rank one operators {Ri} U such that (1) Let N and U be as in (6), and T ∈ U be a trace class operator then there existrank one operators {Ri}U such that T = andAs a corollary, we recapture the distance formula from an arbitrary operator to U.(8) Let A be an atomic Boolean lattice algebra acting on a Hilbert space H, and T be a rank n operator belonging to A, the preannihilator of A. Then T can be written as a finite sum of n rank one operators each belonging to A.In Chapter 4, the vector interpolation problems and operator interpolation problems for a weakly closed nest algebra AlgN-module are studied. The main result is(9) Let N be a nest on a Hilbert space H, and U be a weakly closed AlgAN-module. For X, Y B(H), the following are equivalent:i) There exists an operator T U such that TX = Y ; Moreover, if these conditions are satisfied, then T can be chosen such that In Chapter 5, we deal with the weak topology density of rank one operators in operator subspaces, the following three results are obtained.(10) Let M be a weakly closed operator subspaces acting on a Hilbert space H. M. is said to have Property (P), if the subspace linearly generated by all rank one operators in M is weakly dense in M . We prove that M has Property (P) if and only if(11) Let M be as in (10) then M and every weakly closed subspace including M have Proverty (P) if and only if for any T B(H}, there exist x, y H. such that (12) Let A be a completely distributive subspace lattice algebra acting on a Hilbert space, then the rank one subalgebra of A is dense in A if and only if, the weak closures of the first and the second preannihilators of A in the space of all trace class operators are reflexive.In Chapter 6, we mainly discuss the multiplicative mappings of certain non-standard operator algebras. We prove the following(13) Let H1,H2,H3 ..... be a sequence of complex separable Hilbert spaces, denoteand A A--> A is a multiplicative surjectivemapping which preserves operation and is spectrum-preserving (no linearity or conti-nuity is assumed), then (A) = UAU for all A A, where U : is a linear unitary operator.In Chapter 7, we investigate the commutants of some re...