-class and its bicommutant Vladimir Strauss 2nd Najman Conference - - PowerPoint PPT Presentation

2nd Najman Conference

On a commutative WJ∗-algebra of D+

κ -class and

its bicommutant

Vladimir Strauss

2nd Najman Conference

Introduction

WJ∗-algebra is a weakly closed symmetric (according to the inner product) algebra of operators in a Krein space that contains the

• identity. An operator family belongs to the D+

κ -class if it has at

least one common invariant subspace that is a maximal non-negative subspace and can be presented as a direct sum of its κ-dimensional isotropic part and a uniformly positive subspace, κ < ∞. Let us note that every commutative operator family of self-adjoint operators in Pontryagin spaces belongs to D+

κ -class for

some κ. Finally, the bicommutant of an operator family is the algebra of operators that commute with every operator which commutes with all operators of the given family. We’ll discuss the relation between a function representation for a commutative WJ∗-algebra of D+

κ -class and its bicommutant.

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Introduction (cont.)

A well-known theorem of J. von Neumann says that the bicommutant of an arbitrary W ∗-algebra in a separable Hilbert space coincides with the algebra. If we replace a W ∗-algebra by a WJ∗-algebra, the corresponding result is false even for a finite-dimensional Pontryagin space with the index of indefiniteness equal one (i.e. for a finite-dimensional space Π1). If we consider

• nly commutative WJ∗-algebras, then for the Pontryagin space Π1

(including infinite-dimensional case) an analog of J. von Neumann’s Theorem is true, but this result cannot be extended even for the case of the space Π2. On the other hand in the case of the space Π2 the bicommutant of a commutative WJ∗-algebra can be wider that the initial algebra only on account of some nilpotent operators and not on account of operators with a non-trivial spectral part. We show that the latter result cannot be extended for algebras in Πκ with a big κ and study the corresponding properties of a commutative WJ∗-algebra of the mentioned above class.

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Introduction (cont.)

The symbol Alg A means the minimal weakly closed algebra with the identity that contains an operator A. If Y is an operator family then the symbol Y′ refers to the commutant of Y, i.e. to the algebra of all operators B such that AB = BA for every A ∈ Y. The algebra Y′′ = (Y′)′ is said to be a bicommutant of Y. An algebra A is called reflexive if A′′ = A.

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Examples

Example 1 Let us consider the algebra of 2 × 2 complex triangular matrices A = α β γ . It is easy to see that A is a non-commutative WJ∗-algebra with J = 1 1

• . The direct calculation brings

A′ = ν ν , so A′′ = A.

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Examples (cont.)

Example 2 Assume that the space H is formed by an orthonormalized basis {ej}4

1, the fundamental symmetry J is given by the equalities

Je0 = e1, Je1 = e0, Je2 = e3, Je3 = e2, and a WJ∗-algebra A is generated by the identical operator and the following operators A1: A1e0 = e2, A1e1 = 0, A1e2 = 0, A1e3 = e1 ; A2: A2e0 = ie2, A2e1 = 0, A2e2 = 0, A2e3 = −ie1 ; S: Se0 = e1, Se1 = Se2 = Se3 = 0. Note that the operators A1, A2 and S are J-Js.a., A2

1 = A2 2 = A1A2 = A1S = A2S = S2 = 0.

It is easy to show that A′ is spanned by A and A3: A3e0 = 0, A3e1 = 0, A3e2 = 0, A3e3 = e2. The algebra A′ is commutative and A′′= A′.

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History

Theorem 1. Let A be a J-self-adjoint operator in a separable Pontryagin space of type Π1 and let A = Alg A. Then A′′ = A. V.A.Strauss, On a function calculus for π-self-adjoint operators. 1X School on Operator Theory in Function Spaces (1984), Book of Abstracts, Ternopol, Ukraine, 151–152 (Russian). V.A. Strauss, Functional representation of operators that doubly commute with a selfadjoint operator in a Pontryagin space. (Russian) Sibirsk. Mat. Zh. 29 (1988), no. 6, 176–184; translation in Siberian Math. J. 29 (1988), no. 6, 1012–1018 (1989).

• V. Strauss, A model representation for a simplest -selfadjoint
• perator. In Collection: Funktsionalnii Analiz, Spectral Theory,

State Pedagogical Institute of Uliyanovsk, Ulyanovsk (1984) 123-133. (Russian)

• V. Strauss, Models of Function Type for Commutative Symmetric

Operator Families in Krein Spaces. Abstract and Applied Analysis 2008 (2008), Article ID 439781, 40 pp.

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History (cont.)

Theorem 2. Let A be a commutative WJ∗-algebra in a separable Pontryagin space of type Π1. Then A′′ = A.

• S. N. Litvinov, Description of commutative symmetric algebras in

the Pontryagin space Π1. DAN UzSSR (1987) No 1, 9–12 (Russian). O.Ya. Bendersky, S. N. Litvinov and V. I. Chilin, A description of commutative symmetric operator algebras in a Pontryagin space Π1. Preprint, Tashkent 1989 (Russian). O.Ya. Bendersky, S. N. Litvinov and V. I. Chilin, A description of commutative symmetric operator algebras in a Pontryagin space Π1. Journal of Operator Theory 37 (1997) Issue 2, pp. 201-222. V.S. Shulman, Symmetric Banach Algebras in spaces of type Π1. Mat.sb. 89 (1972) No 2, 264–279 (Russian).

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Next step

A subspace L is called pseudoregular if it can be presented in the form L = ˆ L ∔ L1, where ˆ L is a regular subspace and L1 is an isotropic part of L (i.e., L1 = L ∩ L[⊥]). A J-symmetric operator family Y belongs to the class D+

κ if there

is a subspace L+ in H, such that

◮ L+ is Y-invariant, ◮ the subspace L+ is simultaneously maximal non-negative and

pseudoregular,

◮ dim(L+ ∩ L[⊥] + ) = κ.

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Next step (cont.)

Theorem 3 Let Y ∈ D+

κ be a commutative family of J-s.a.

• perators with real spectra. Then there exists a J-orth.sp.f. E with

a finite peculiar spectral set Λ, such that the following conditions hold

◮ Eλ ∈ Alg Y for all λ ∈ R\Λ, ◮ ∀ A ∈ Y and for every closed interval ∆ ⊂ R\Λ the operator

AE(∆) is spectral,

◮ ∀ A ∈ Y, ∃ a defined almost everywhere function φ(λ), such

that for every closed interval ∆ ⊂ R\Λ the decomposition AE(∆) =

• ∆ φ(λ)E(dλ) is valid.
• T. Ya. Azizov, V. A. Strauss, Spectral decompositions for special

classes of self-adjoint and normal operators on Krein spaces. Spectral Theory and its Applications, Proceedings dedicated to the 70-th birthday of Prof. I.Colojoar˘ a, Theta 2003, 45–67.

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Next step (cont.)

Let ϕ(t) be a continuous scalar function vanishing near Λ. Set Bϕ = 1

−1 ϕ(t)dEλ, where the improper integral has the obvious

• meaning. Let AΛ be the weak closure of the operator set {Bϕ}

generated by the latter representation. Theorem 4 For the given algebra A ∈ D+

κ there is a finite

collection of J-s.a. operators A1, A2, . . . , Al ∈ A, such that every

• perator B ∈ A has a representation

B = C + F + Q(A1, A2, . . . , Al), where C ∈ A is a nilpotent

• perator, F ∈ AΛ and Q(t1, t2, . . . , tl) is a polynomial of l

variables.

• V. Strauss, A functional description for the commutative

WJ∗-algebras of the D+

κ -class. Proceedings of Colloquium on

Operator Theory and its Applications dedicated to Prof. Heinz Langer (Vienna, 2004), in Operator Theory: Advances and Applications 163 (2005), Birkh¨ auser Verlag, 299–335.

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Next step (cont.)

Theorem 5 For the given algebra A ∈ D+

κ there are operators

B1, B2, . . . , Bn ∈ A′′, such that for every operator B ∈ A′′ the representation B = Z + F + Q(B1, B2, . . . , Bn) + C holds. Here Z ∈ A′′ is a nilpotent operator, F ∈ AΛ and Q(ξ1, ξ2, . . . , ξn) is a polynomial of n variables. Remark Let A0 and (A′′)0 be the nilpotent part of, respectively, algebras A and A′′. Then, generally speaking, in the latter representation A0 = (A′′)0 and l = n.