Contemporary Mathematics. Fundamental Directions, 2018
In this paper, we review main directions and results of the theory of topological radicals. We co... more In this paper, we review main directions and results of the theory of topological radicals. We consider applications to different problems in the theory of operators and Banach algebras.
It came as a bit of a shock to all of us: Victor (or Vitya, as we affectionately call him in Russ... more It came as a bit of a shock to all of us: Victor (or Vitya, as we affectionately call him in Russian) is 65. Hold on, this can’t be true! We just played football with him and he scored a few goals. He was all over the field – full of drive and the desire to win. No, he cannot be 65! Don’t tell us about his date of birth or the age of his children.
ABSTRACT. In this paper we study "scalar" and "operator" smoothness condition... more ABSTRACT. In this paper we study "scalar" and "operator" smoothness conditions for functions to act on symmetrically normed ideals of B(H) and on the domains of weakly* closed derivations of these ideals.
Spectral criteria for the cohomological triviality of extensions of representations of connected ... more Spectral criteria for the cohomological triviality of extensions of representations of connected nilpotent groups are obtained. They are applied to the study of symmetrized extensions of unitary representations by finite-dimensional representations and to the theory of J-unitary representations of groups on Pontryagin spaces.
Let λ be a finite-dimensional representation of a connected nilpotent group G and U be a unitary ... more Let λ be a finite-dimensional representation of a connected nilpotent group G and U be a unitary representation of G. We investigate the structure of the extensions of λ by U and, correspondingly, the group H 1 (λ, U) of 1-cohomologies. A spectral criterion of triviality of H 1 (λ, U) is proved and systematically used in the study of various types of decomposition of the extensions. We consider a special type of (λ, U)-cocycles – neutral cocycles, which play a crucial role in the theory of J-unitary representations of groups on Pontryagin Π k-spaces.
In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it t... more In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codimensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via subspace-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of closed Lie subalgebras of finite codimension (closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) produce different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained. Response to Reviewers: We are very grateful to the reviewer for his perceptive comments Abstract. In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codi-mensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via sub-space-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of closed Lie subal-gebras of finite codimension (closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) produce different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained.
This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, 'Banach ... more This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, 'Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals', J. Funct. Anal. 256 (2009) 323–351.] and is devoted to the study of reducibility of an infinite-dimensional Lie algebra of operators on a Banach space when its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. In addition to the tools developed in the above paper; filtrations of Banach spaces with respect to Lie algebras of operators and related systems of operators on graded Banach spaces, the present paper introduces and studies some new concepts and techniques: the theory of Lie quasi-ideals and properties of Lie nilpotent finite-dimensional subspaces of Banach associative algebras. The application of these techniques to an operator Lie algebra L shows that, under some mild additional assumptions, L is reducible if its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. This, in turn, leads to the main result of the paper: if a Banach Lie algebra L has a closed Lie subalgebra of finite codimension, then it has a proper closed Lie ideal of finite codimension. Moreover, if L is non-commutative, then it has a characteristic Lie ideal of finite codimension, that is, a proper closed Lie ideal of L invariant for all bounded derivations of L.
It is proved that, for any Lipschitz function f (t 1 ,. .. , t n) of n variables, the correspondi... more It is proved that, for any Lipschitz function f (t 1 ,. .. , t n) of n variables, the corresponding map f op : (A 1 ,. .. , A n) → f (A 1 ,. .. , A n) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S p , p ∈ (1, ∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S p. It is also proved that the map f op is Fréchet differentiable in the norm of S p if f is continuously differentiable.
i esex weriwesgev ty xev sx PHUUEWVUW olume QD xumer I @PHIPAD PW ! RH wsxswe gyxhssyx... more i esex weriwesgev ty xev sx PHUUEWVUW olume QD xumer I @PHIPAD PW ! RH wsxswe gyxhssyx py greix shiev yp gyweg yiey F pormisnoD i uissin gommunited y FF rrykov uey wordsX ompt opertorsD htten idelsD minimx onditionsF ew wthemtis ujet glssi(tionX RUfIHD RWtQSF estrtF sn this pper we study the vlidity of vrious types of minimx ondition for opertors in htten idels of ompt opertors on seprle rilert spesF I sntrodution vet f : X × Λ → R e rel funtion on the produt of nonEempty sets X nd Λ. he lssil minimx prolem is the prolem to (nd suitle onditions on f tht gurntee the vlidity of the equlity inf x∈X sup λ∈Λ f (x, λ) = sup λ∈Λ inf x∈X f (x, λ). @IFIA por n overview of the sujet see UF forenshtein nd hulmn proved in R tht if X is ompt metri speD Λ is rel intervl nd f is ontinuousD then @IFIA holds provided thtD for eh x ∈ X, the funtion f (x, ·) is onveY nd for eh λ ∈ Λ, every lol minimum of the funtion f (·, λ) is glol minimumF ome weker onditions on f tht ensure the vlidity of @IFIA were estlished y int ymond in W nd ieri in VF he minimx onditions hs mny interesting pplitions to the opertor theory @see ID PD R nd TAF sn RD for exmpleD the uthors used @IFIA to prove esplundE tk equlity inf λ∈C A − λB = sup x∈B(H) inf λ∈C Ax − λBx , for opertors A, B on rilert spe H nd estlished thtD for fnh spesD it should e repled y n inequlityF sn this pper we study the vlidity of vrious types of minimx ondition @IFIA for opertors in htten idels of ompt opertorsF hese minimx onditions re linked to the pproximtion of opertors y (niteErnk opertors in the htten normsF vet H e seprle rilert speF vet B(H) e the gBElger of ll ounded opertors on H with opertor norm ·· nd let C(H) e the losed idel of ll ompt opertors in B(H)F e twoEsided idel J of B(H) is symmetrilly normed @sF nFA if @see SA it is fnh spe in its own norm · J nd AXB J ≤ AX J B, for A, B ∈ B(H) nd X ∈ J. fy glkin theoremD ll sF nF idels lie in C(H).
The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded oper... more The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L 0 that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L 0 has finite codimension in L and there are Lie subalgebras L 0 = L 0 ⊂ L 1 ⊂ · · · ⊂ L p = L such that L i+1 = L i + [L i , L i+1 ] for all i; (2) L 0 is a Lie ideal of L and dim(L 0) = ∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.
Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submo... more Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submodule of X if Ay + yA ∈ Y for all A ∈ A and y ∈ Y (if X = A, then this coincides with the classical concept of a Jordan ideal). When is a Jordan A-submodule a submodule? We give a thorough analysis of this question in both algebraic and analytic context. In the first part of the paper, we consider general algebras and general Banach algebras. In the second part, we treat some more specific topics, such as symmetrically normed Jordan A-submodules. Some of our results are of interest also in the classical situation; in particular, we show that there exist C *-algebras having Jordan ideals that are not ideals.
Let A be an algebra, and let X be an arbitrary A-bimodule. A linear space Y ⊂ X is called a Jorda... more Let A be an algebra, and let X be an arbitrary A-bimodule. A linear space Y ⊂ X is called a Jordan A-submodule if Ay + yA ∈ Y for all A ∈ A and y ∈ Y. (For X = A , this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C *-algebras. It turns out that there exist C *-algebras in which not all Jordan ideals are ideals.
Contemporary Mathematics. Fundamental Directions, 2018
In this paper, we review main directions and results of the theory of topological radicals. We co... more In this paper, we review main directions and results of the theory of topological radicals. We consider applications to different problems in the theory of operators and Banach algebras.
It came as a bit of a shock to all of us: Victor (or Vitya, as we affectionately call him in Russ... more It came as a bit of a shock to all of us: Victor (or Vitya, as we affectionately call him in Russian) is 65. Hold on, this can’t be true! We just played football with him and he scored a few goals. He was all over the field – full of drive and the desire to win. No, he cannot be 65! Don’t tell us about his date of birth or the age of his children.
ABSTRACT. In this paper we study "scalar" and "operator" smoothness condition... more ABSTRACT. In this paper we study "scalar" and "operator" smoothness conditions for functions to act on symmetrically normed ideals of B(H) and on the domains of weakly* closed derivations of these ideals.
Spectral criteria for the cohomological triviality of extensions of representations of connected ... more Spectral criteria for the cohomological triviality of extensions of representations of connected nilpotent groups are obtained. They are applied to the study of symmetrized extensions of unitary representations by finite-dimensional representations and to the theory of J-unitary representations of groups on Pontryagin spaces.
Let λ be a finite-dimensional representation of a connected nilpotent group G and U be a unitary ... more Let λ be a finite-dimensional representation of a connected nilpotent group G and U be a unitary representation of G. We investigate the structure of the extensions of λ by U and, correspondingly, the group H 1 (λ, U) of 1-cohomologies. A spectral criterion of triviality of H 1 (λ, U) is proved and systematically used in the study of various types of decomposition of the extensions. We consider a special type of (λ, U)-cocycles – neutral cocycles, which play a crucial role in the theory of J-unitary representations of groups on Pontryagin Π k-spaces.
In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it t... more In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codimensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via subspace-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of closed Lie subalgebras of finite codimension (closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) produce different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained. Response to Reviewers: We are very grateful to the reviewer for his perceptive comments Abstract. In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codi-mensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via sub-space-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of closed Lie subal-gebras of finite codimension (closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) produce different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained.
This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, 'Banach ... more This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, 'Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals', J. Funct. Anal. 256 (2009) 323–351.] and is devoted to the study of reducibility of an infinite-dimensional Lie algebra of operators on a Banach space when its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. In addition to the tools developed in the above paper; filtrations of Banach spaces with respect to Lie algebras of operators and related systems of operators on graded Banach spaces, the present paper introduces and studies some new concepts and techniques: the theory of Lie quasi-ideals and properties of Lie nilpotent finite-dimensional subspaces of Banach associative algebras. The application of these techniques to an operator Lie algebra L shows that, under some mild additional assumptions, L is reducible if its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. This, in turn, leads to the main result of the paper: if a Banach Lie algebra L has a closed Lie subalgebra of finite codimension, then it has a proper closed Lie ideal of finite codimension. Moreover, if L is non-commutative, then it has a characteristic Lie ideal of finite codimension, that is, a proper closed Lie ideal of L invariant for all bounded derivations of L.
It is proved that, for any Lipschitz function f (t 1 ,. .. , t n) of n variables, the correspondi... more It is proved that, for any Lipschitz function f (t 1 ,. .. , t n) of n variables, the corresponding map f op : (A 1 ,. .. , A n) → f (A 1 ,. .. , A n) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S p , p ∈ (1, ∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S p. It is also proved that the map f op is Fréchet differentiable in the norm of S p if f is continuously differentiable.
i esex weriwesgev ty xev sx PHUUEWVUW olume QD xumer I @PHIPAD PW ! RH wsxswe gyxhssyx... more i esex weriwesgev ty xev sx PHUUEWVUW olume QD xumer I @PHIPAD PW ! RH wsxswe gyxhssyx py greix shiev yp gyweg yiey F pormisnoD i uissin gommunited y FF rrykov uey wordsX ompt opertorsD htten idelsD minimx onditionsF ew wthemtis ujet glssi(tionX RUfIHD RWtQSF estrtF sn this pper we study the vlidity of vrious types of minimx ondition for opertors in htten idels of ompt opertors on seprle rilert spesF I sntrodution vet f : X × Λ → R e rel funtion on the produt of nonEempty sets X nd Λ. he lssil minimx prolem is the prolem to (nd suitle onditions on f tht gurntee the vlidity of the equlity inf x∈X sup λ∈Λ f (x, λ) = sup λ∈Λ inf x∈X f (x, λ). @IFIA por n overview of the sujet see UF forenshtein nd hulmn proved in R tht if X is ompt metri speD Λ is rel intervl nd f is ontinuousD then @IFIA holds provided thtD for eh x ∈ X, the funtion f (x, ·) is onveY nd for eh λ ∈ Λ, every lol minimum of the funtion f (·, λ) is glol minimumF ome weker onditions on f tht ensure the vlidity of @IFIA were estlished y int ymond in W nd ieri in VF he minimx onditions hs mny interesting pplitions to the opertor theory @see ID PD R nd TAF sn RD for exmpleD the uthors used @IFIA to prove esplundE tk equlity inf λ∈C A − λB = sup x∈B(H) inf λ∈C Ax − λBx , for opertors A, B on rilert spe H nd estlished thtD for fnh spesD it should e repled y n inequlityF sn this pper we study the vlidity of vrious types of minimx ondition @IFIA for opertors in htten idels of ompt opertorsF hese minimx onditions re linked to the pproximtion of opertors y (niteErnk opertors in the htten normsF vet H e seprle rilert speF vet B(H) e the gBElger of ll ounded opertors on H with opertor norm ·· nd let C(H) e the losed idel of ll ompt opertors in B(H)F e twoEsided idel J of B(H) is symmetrilly normed @sF nFA if @see SA it is fnh spe in its own norm · J nd AXB J ≤ AX J B, for A, B ∈ B(H) nd X ∈ J. fy glkin theoremD ll sF nF idels lie in C(H).
The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded oper... more The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L 0 that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L 0 has finite codimension in L and there are Lie subalgebras L 0 = L 0 ⊂ L 1 ⊂ · · · ⊂ L p = L such that L i+1 = L i + [L i , L i+1 ] for all i; (2) L 0 is a Lie ideal of L and dim(L 0) = ∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.
Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submo... more Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submodule of X if Ay + yA ∈ Y for all A ∈ A and y ∈ Y (if X = A, then this coincides with the classical concept of a Jordan ideal). When is a Jordan A-submodule a submodule? We give a thorough analysis of this question in both algebraic and analytic context. In the first part of the paper, we consider general algebras and general Banach algebras. In the second part, we treat some more specific topics, such as symmetrically normed Jordan A-submodules. Some of our results are of interest also in the classical situation; in particular, we show that there exist C *-algebras having Jordan ideals that are not ideals.
Let A be an algebra, and let X be an arbitrary A-bimodule. A linear space Y ⊂ X is called a Jorda... more Let A be an algebra, and let X be an arbitrary A-bimodule. A linear space Y ⊂ X is called a Jordan A-submodule if Ay + yA ∈ Y for all A ∈ A and y ∈ Y. (For X = A , this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C *-algebras. It turns out that there exist C *-algebras in which not all Jordan ideals are ideals.
Uploads
Papers