Continuity of LF-algebra representations associated to representations of Lie groups

Glöckner, Helge

doi:10.1215/21562261-2265895

Cited by 6 publications

(5 citation statements)

References 36 publications

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“…Every Banach representation is both a projective and an inductive generalized proto-Banach representation. Furthermore, every proto-Banach representation [15] (and thus particularly every F -representation [14]) is a projective generalized proto-Banach representation.…”

Section: Ultradifferentiable Vectors and The Factorization Theoremmentioning

confidence: 99%

“…The main goal of this article is to prove this conjecture for G = (R d , +) and further extend it in two directions. On the one hand, we consider two types of moderate growth representations (π, E) of (R d , +) on sequentially complete locally convex Hausdorff spaces E (including proto-Banach representations [15] and thus F -representations). This setting is indispensable for the applications we have in mind.…”

Section: Introductionmentioning

confidence: 99%

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Factorization in Denjoy-Carleman classes associated to representations of (Rd,+)

Debrouwere

Prangoski

Vindas

2021

Journal of Functional Analysis

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For two types of moderate growth representations of (R d , +) on sequentially complete locally convex Hausdorff spaces (including F-representations [14]), we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [14, Conjecture 6.4] for analytic vectors of representations of G = (R d , +). As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property.

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Section: Ultradifferentiable Vectors and The Factorization Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Factorization in Denjoy-Carleman classes associated to representations of (Rd,+)

Debrouwere

Prangoski

Vindas

2021

Journal of Functional Analysis

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Section: Ultradifferentiable Vectors and The Factorization Theoremmentioning

confidence: 99%

Factorization in Denjoy-Carleman classes associated to representations of $(\mathbb{R}^{d},+)$

Debrouwere,

Prangoski,

Vindas

2020

Preprint

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“…Theorem 9.2] combined with the Chain Rule 3. 19. Notice however that one still needs f to be a C r,k -mapping with k ≥ r + s, as also the Chain Rule 3.19 decreases the order of differentiability.…”

Section: Parameter Dependent Differential Equationsmentioning

confidence: 99%

Differentiable mappings on products with different degrees of differentiability in the two factors

Alzaareer

Schmeding

2015

Expositiones Mathematicae

120

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We develop differential calculus of C r,s -mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class C r,s .MSC 2000 Subject Classification: Primary 26E15; Secondary 26E20, 46E25, 34A12, 22E65, 46T20 (A. Schmeding) spaces, U ⊆ E 1 and V ⊆ E 2 be open subsets and r, s ∈ N 0 ∪ {∞}. We say that a map f :exist for all for all i, j ∈ N 0 with i ≤ r and j ≤ s, and are continuous functions in (x, y, w 1 , . . . , w i , v 1 , . . . , v j ) ∈ U ×V ×E i 1 ×E j 2 (see Definition 3.1 for details). To enable choices like U = [0, 1], and also with a view towards manifolds with boundary, more generally we consider C r,s -maps if U and V are locally convex (in the sense that each point has a convex neighbourhood) and have dense interior (see Definition 3.2). These properties are satisfied by all open sets. Variants and special cases of C r,s -mappings are encountered in many parts of analysis. For example, [2] considers analogues of C 0,r -maps on Banach spaces based on continuous Fréchet differentiability; see [12, 1.4] for C 0,r -maps; [14] for C r,s -maps on finite-dimensional domains; and [11, p. 135] for certain Lip r,s -maps in the convenient setting of analysis. Cf. also [32], [20] for ultrametric analogues in finite dimensions. Furthermore, a key result concerning C r,s -maps was conjectured in [15, p.10]. However the authors' interest concerning the subject was motivated by recent questions in infinite dimensional Lie theory. At the end of this section, we present an overview, showing where refined tools from C r,s -calculus are useful. The first aim of this paper is to develop necessary tools like a version of the Theorem of Schwarz and various versions of the Chain Rule. After that we turn to an advanced tool, the exponential law for spaces of mappings on products (Theorem 3.28). We endow spaces of C r -maps with the usual compact-open C r -topology (as recalled in Definition 2.5) and spaces of C r,s -maps with the analogous compact-open C r,stopology (see Definitions 3.21 and 4.3). Recall that a topological space X is called a k-space if it is Hausdorff and its topology is the final topology with respect to the inclusion maps K → X of compact subsets of X (cf.[26] and the work [39], which popularized the use of k-spaces in algebraic topology). For example, all locally compact spaces and all metrizable topological spaces are k-spaces. The main results of Section 3 (Theorems 3.25 and 3.28) subsume:Theorem A. Let E 1 , E 2 and F be locally convex spaces, U ⊆ E 1 and V ⊆ E 2 be locally convex subsets with dense interior, and r, s ∈ N 0 ∪ {∞}. Then γ ∨ : U → C s (V, F ), x → γ(x, •) is C r for each γ ∈ C r,s (U × V, F ), and the map Φ : C r,s (U × V, F ) → C r (U, C s (V, F )), γ → γ ∨ (1)is linear and a topological embedding. If U × V × E 1 × E 2 is a k-space or V is locally compact, then Φ is...

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Continuity of LF-algebra representations associated to representations of Lie groups

Cited by 6 publications

References 36 publications

Factorization in Denjoy-Carleman classes associated to representations of (Rd,+)

Factorization in Denjoy-Carleman classes associated to representations of (Rd,+)

Factorization in Denjoy-Carleman classes associated to representations of $(\mathbb{R}^{d},+)$

Differentiable mappings on products with different degrees of differentiability in the two factors

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