Quasispectra of solvable Lie algebra homomorphisms into Banach algebras

Dosiev, Anar

doi:10.4064/sm174-1-2

Cited by 5 publications

(7 citation statements)

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“…The algebra A (p) is called a Banach enveloping algebra of the Banach Lie algebra g with the norm p (see [5] and [8]). If g is a nilpotent Lie algebra, then A (p) is a norm-completion of U (g) (see [3] and [5]).…”

Section: The Fréchet Algebrasmentioning

confidence: 99%

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Formally-radical Functions in Elements of a Nilpotent Lie Algebra and Noncommutative Localizations

Dosi

2010

Algebra Colloq.

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In the present paper, we introduce the sheaf Tg of germs of non-commutative holomorphic functions in elements of a finite-dimensional nilpotent Lie algebra g, which is a sheaf of non-commutative Fréchet algebras over the character space of g. We prove that Tg(D) is a localization over the universal enveloping algebra U (g) whenever D is a polydisk, which in turn allows to describe the Taylor spectrum of a supernilpotent Lie algebra of operators in terms of the transversality. 2000 Mathematics Subject Classification: primary 46H30, 47A60; secondary 46M18, 16L30, 16S30, 18G25 Keywords: non-commutative holomorphic functions in elements of a Lie algebra, formallyradical functions, non-commutative localization, Taylor spectrum, transversality Algebra Colloq. 2010.17:749-788. Downloaded from www.worldscientific.com by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 08/15/14. For personal use only. PreliminariesAll considered linear spaces are complex and algebras are assumed to be unital and associative. Given a linear space X, ∧X = k≥0 ∧ k X is the exterior algebra of X. If u = u 1 ∧ · · · ∧ u k ∈ ∧ k X is a k-vector, then we use the following denotation u i = u 1 ∧ · · · ∧ u i ∧ · · · ∧ u k for the (k − 1)-vector, where u i means the omission of the variable u i . If we throw out two variables u i and u j , i < j, from the expression of u, the obtained vector is denoted by u ij . The space of all X-valued polynomials in s variables is denoted by X[ω 1 , . . . , ω s ]. Suppose X is a Fréchet space with its defining countable seminorm set {p t : t ∈ Λ}. Let X [[ω 1 , . . . , ω s ]] be the space of all X-valued formal power series in s variables, so each its element f has a unique formal power series expansion f = J∈Z s1 · · · ω js s . Algebra Colloq. 2010.17:749-788. Downloaded from www.worldscientific.com by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 08/15/14. For personal use only. 752 A. DosiThe space X [[ω 1 , . . . , ω s ]] turns into a Fréchet space with respect to a seminorm set q t,K : (t, K) ∈ Λ × Z s + , where q t,K (f ) = max {p t (x J ) : J ≤ K}. One may easily verify that the topology generated by the latter seminorm set is merely the direct product topology of X Z s + . In particular,and if X is a nuclear space, then X [[ω 1 , . . . , ω s ]] is a Fréchet nuclear space. If X and Y are Hausdorff locally convex spaces, then the space of all continuous linear mappings X → Y is denoted by L (X, Y ), we also write L(X) instead of L (X, X). We use the conventional denotation X ⊗ Y for the projective tensor product. An element T ∈ L (X, Y ) is said to be a (co)retraction if T S = 1 (respectively, ST = 1) for a certain S ∈ L (Y, X). In this case, Y is called a (co)retract of X. The Jacobson radical of an algebra A is denoted by Rad A. The left (respectively, right) multiplication operator on A is denoted by L a (respectively, R a ), that is,

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Section: The Fréchet Algebrasmentioning

confidence: 99%

“…Some properties of the algebras O g (U ) on absolutely convex domains U including the relevant functional calculus were investigated in [5] and [8]. How extend these approach and results when the domain U is arbitrary, not necessarily absolutely convex?…”

Section: Introductionmentioning

confidence: 99%

Formally-radical Functions in Elements of a Nilpotent Lie Algebra and Noncommutative Localizations

Dosi

2010

Algebra Colloq.

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“…[8]- [12]) спектральной теории нильпотентных операторных алгебр Ли открывают возможности применения метода Тейлора к универсальной обертывающей алгебре B = U (g) конечномерной нильпотент-ной алгебры Ли g. В частности, можно дать определение спектра Тейлора σ(g, X) банахова U (g)-модуля X, обладающего свойством спектрального отоб-ражения по отношению к некоммутативным полиномам.…”

Section: § 1 введениеunclassified

Спектр Тейлора И Трансверсальность Для Операторной Алгебры Гейзенберга

Dosi¹,

Dosi²

2010

Матем. сб.

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Спектр Тейлора и трансверсальность для операторной алгебры ГейзенбергаРассмотрены проблемы некоммутативного голоморфного функцио-нального исчисления для банаховых модулей над конечномерными ниль-потентными алгебрами Ли. Основной результат устанавливает свойство трансверсальности алгебр некоммутативных голоморфных функций по отношению к спектру Тейлора семейства ограниченных линейных опера-торов, порождающего алгебру Гейзенберга.Библиография: 25 названий.Ключевые слова: голоморфная функция от элементов алгебры Ли, спектр Тейлора, свойство трансверсальности, обращающее пополнение Фреше.

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“…The class of noncommutative supernilpotent Lie algebras of operators on an infinite-dimensional Banach space X is sufficiently wider than the class of commutative Lie algebras. The well developed joint spectral theory for an operator tuple T generating a nilpotent Lie algebra has been proposed in papers [1], [6], [7], [12]. In particular, we have a well defined Taylor spectrum σ (T ) of the operator tuple T which possesses the spectral mapping property with respect to the noncommutative polynomials.…”

Section: Introductionmentioning

confidence: 99%