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. 2nd Najman Conference
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 Π 1 ). If we consider only commutative WJ ∗ -algebras, then for the Pontryagin space Π 1 (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 Π 2 . On the other hand in the case of the space Π 2 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 Π κ with a big κ and study the corresponding properties of a commutative WJ ∗ -algebra of the mentioned above class. 2nd Najman Conference
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 . 2nd Najman Conference
Examples Example 1 Let us consider the algebra of 2 × 2 complex triangular matrices � � α 0 � � A = . It is easy to see that A is a non-commutative β γ � 0 1 � WJ ∗ -algebra with J = . The direct calculation brings 1 0 � � ν � � 0 A ′ = , so A ′′ � = A . 0 ν 2nd Najman Conference
Examples (cont.) Example 2 Assume that the space H is formed by an orthonormalized basis { e j } 4 1 , the fundamental symmetry J is given by the equalities Je 0 = e 1 , Je 1 = e 0 , Je 2 = e 3 , Je 3 = e 2 , and a WJ ∗ -algebra A is generated by the identical operator and the following operators A 1 : A 1 e 0 = e 2 , A 1 e 1 = 0 , A 1 e 2 = 0 , A 1 e 3 = e 1 ; A 2 : A 2 e 0 = ie 2 , A 2 e 1 = 0 , A 2 e 2 = 0 , A 2 e 3 = − ie 1 ; S : Se 0 = e 1 , Se 1 = Se 2 = Se 3 = 0 . Note that the operators A 1 , A 2 and S are J -Js.a., 2 = A 1 A 2 = A 1 S = A 2 S = S 2 = 0. A 2 1 = A 2 It is easy to show that A ′ is spanned by A and A 3 : A 3 e 0 = 0 , A 3 e 1 = 0 , A 3 e 2 = 0 , A 3 e 3 = e 2 . The algebra A ′ is commutative and A ′′ = A ′ . 2nd Najman Conference
History Theorem 1 . Let A be a J -self-adjoint operator in a separable Pontryagin space of type Π 1 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 operator. 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. 2nd Najman Conference
History (cont.) Theorem 2 . Let A be a commutative WJ ∗ -algebra in a separable Pontryagin space of type Π 1 . Then A ′′ = A . S. N. Litvinov, Description of commutative symmetric algebras in the Pontryagin space Π 1 . 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 Π 1 . 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 Π 1 . Journal of Operator Theory 37 (1997) Issue 2, pp. 201-222. V.S. Shulman, Symmetric Banach Algebras in spaces of type Π 1 . Mat.sb. 89 (1972) No 2, 264–279 (Russian). 2nd Najman Conference
Next step A subspace L is called pseudoregular if it can be presented in the form L = ˆ L ∔ L 1 , where ˆ L is a regular subspace and L 1 is an isotropic part of L (i.e., L 1 = 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 . 2nd Najman Conference
Next step (cont.) Theorem 3 Let Y ∈ D + κ be a commutative family of J -s.a. operators 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 ◮ E λ ∈ 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 ( d λ ) 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. 2nd Najman Conference
Next step (cont.) Let ϕ ( t ) be a continuous scalar function vanishing near Λ. Set � 1 B ϕ = − 1 ϕ ( t ) dE λ , where the improper integral has the obvious meaning. Let A Λ be the weak closure of the operator set { B ϕ } generated by the latter representation. Theorem 4 For the given algebra A ∈ D + κ there is a finite collection of J -s.a. operators A 1 , A 2 , . . . , A l ∈ A , such that every operator B ∈ A has a representation B = C + F + Q ( A 1 , A 2 , . . . , A l ), where C ∈ A is a nilpotent operator, F ∈ A Λ and Q ( t 1 , t 2 , . . . , t l ) 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. 2nd Najman Conference
Next step (cont.) Theorem 5 For the given algebra A ∈ D + κ there are operators B 1 , B 2 , . . . , B n ∈ A ′′ , such that for every operator B ∈ A ′′ the representation B = Z + F + Q ( B 1 , B 2 , . . . , B n ) + C holds. Here Z ∈ A ′′ is a nilpotent operator, F ∈ A Λ and Q ( ξ 1 , ξ 2 , . . . , ξ n ) is a polynomial of n variables. Remark Let A 0 and ( A ′′ ) 0 be the nilpotent part of, respectively, algebras A and A ′′ . Then, generally speaking, in the latter representation A 0 � = ( A ′′ ) 0 and l � = n . 2nd Najman Conference
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