Continuity of Convolution of Test Functions on Lie Groups

Birth, Lidia; Glöckner, Helge

doi:10.4153/cjm-2012-035-6

Cited by 5 publications

(17 citation statements)

References 33 publications

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“…That the convolution product on D(G) is jointly continuous for every Lie group G with at most finitely many connected components has been shown recently by Birth and Glöckner in[BG11].…”

mentioning

confidence: 90%

Reflection positivity and conformal symmetry

Neeb

Ólafsson

2014

Journal of Functional Analysis

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A reflection positive Hilbert space is a triple (E , E+, θ), where E is a Hilbert space, θ a unitary involution and E+ a closed subspace on which the hermitian form v, w θ := θv, w is positive semidefinite. From this data one obtains a Hilbert space E by completing a suitable quotient of E+ with respect to ·, · θ on E+. To obtain compatible unitary representations of Lie groups, we start with triples (G, S, τ ), where G is a Lie group, τ an involutive automorphism of G and S a subsemigroup invariant under the involution) and π(S)E+ ⊆ E+. Motivated by the passage from the euclidean motion group to the Poincaré group in quantum field theory, one expects a duality between reflection positive representations and unitary representations of the dual symmetric Lie group G c on E. We propose a new approach to a reflection positive representations based on reflection positive distributions and reflection positive distribution vectors. In particular, we generalize the Bochner-Schwartz Theorem to positive definite distributions on open convex cones and apply our techniques to complementary series representations of the conformal group O + 1,n+1 (R) of the sphere S n .

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“…That the convolution product on D(G) is jointly continuous for every Lie group G with at most finitely many connected components has been shown recently by Birth and Glöckner in[BG11].…”

mentioning

confidence: 90%

Reflection positivity and conformal symmetry

Neeb

Ólafsson

2014

Journal of Functional Analysis

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“…using the notation from (6). Hence Lemma 4.10 shows that the integral in (18) converges absolutely for all z ∈ xK 0 and defines a C-analytic function xK 0 → E. Notably, this holds for x = x 0 .…”

Section: A Proofs Of the Lemmas In Sections 1 Andmentioning

confidence: 88%

“…It suffices to prove the lemma for r ∈ N 0 . By [6,Lemma A.2], g is continuous. If r > 0, k ∈ N 0 with k ≤ r, p ∈ P and q 1 , .…”

Section: A Proofs Of the Lemmas In Sections 1 Andmentioning

confidence: 99%

“…, t k ) in some open 0-neighbourhood W ⊆ R k . By [6,Lemma A.3], h : W → E is C k , and d (k,0) g(x, p, q 1 , . .…”

Section: A Proofs Of the Lemmas In Sections 1 Andmentioning

confidence: 99%

“…By Lemma 4.14, Π : A(G) × E → E ω is always separately continuous, hypocontinuous in its second argument, and sequentially continuous (hence also the maps in (b) and (c)) 6. (b) follows from (a) as the inclusion E ω → E is continuous linear.…”

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confidence: 94%

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

Glöckner¹

2013

Kyoto J. Math.

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Let G be a finite-dimensional Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and the space E ∞ of smooth vectors are C ∞ c (G)-modules, but the module multiplication need not be continuous. The pathology can be ruled out if E is (or embeds into) a projective limit of Banach G-modules. Moreover, in this case E ω (the space of analytic vectors) is a module for the algebra A(G) of superdecaying analytic functions introduced by Gimperlein, Krötz and Schlichtkrull. We prove that E ω is a topological A(G)-module if E is a Banach space or, more generally, if every countable set of continuous seminorms on E has an upper bound. The same conclusion is obtained if G has a compact Lie algebra. The question of whether C ∞ c (G) and A(G) are topological algebras is also addressed.Classification: 22D15, 46F05 (Primary); 22E30, 42A85, 46A13, 46E25.

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Continuity of bilinear maps on direct sums of topological vector spaces

Glöckner

2012

Journal of Functional Analysis

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We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map f :taking a pair of test functions to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T (E) := j∈N 0 T j (E) as the locally convex direct sum of projective tensor powers of E. We show that T (E) is a topological algebra if and only if every sequence of continuous seminorms on E has an upper bound. In particular, if E is metrizable, then T (E) is a topological algebra if and only if E is normable. Also, T (E) is a topological algebra if E is DFS or k ω . Classification: 46M05 (Primary); 42A85, 44A35, 46A13, 46A11, 46A16, 46E25, 46F05

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Continuity of Convolution of Test Functions on Lie Groups

Abstract: Abstract. For a Lie group G, we show that the map C

Cited by 5 publications

References 33 publications

Reflection positivity and conformal symmetry

Reflection positivity and conformal symmetry

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

Continuity of bilinear maps on direct sums of topological vector spaces

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