Wyniki wyszukiwania

1

Algebras whose groups of units are Lie groups

100%

Glöckner H.

Studia Mathematica

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2002

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tom 153

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nr 2

147-177

EN

Let A be a locally convex, unital topological algebra whose group of units $A^{×}$ is open and such that inversion $ι : A^{×}→ A^{×}$ is continuous. Then inversion is analytic, and thus $A^{×}$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then $A^{×}$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group $A^{×}$ is an analytic Lie group without a globally defined exponential function. We also discuss generalizations in the setting of "convenient differential calculus", and describe various examples.

2

When unit groups of continuous inverse algebras are regular Lie groups

63%

Glöckner H., Neeb K.-H.

Studia Mathematica

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2012

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tom 211

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nr 2

95-109

EN

It is a basic fact in infinite-dimensional Lie theory that the unit group $A^{×}$ of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group $A^{×}$ is regular in Milnor's sense. Notably, $A^{×}$ is regular if A is Mackey-complete and locally m-convex.

3

General Dirichlet series, arithmetic convolution equations and Laplace transforms

51%

Glöckner H., Lucht L., Porubský Š.

Studia Mathematica

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2009

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tom 193

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nr 2

109-129

EN

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_{d}∗g^{∗d} + a_{d-1}∗g^{∗(d-1)} + ⋯ + a₁∗g + a₀ = 0$, where $a₀,...,a_{d}: ℕ → ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form $∑_{x∈X} f(x)e^{-sx}$ ($s ∈ ℂ^{k}$), where $X ⊆ [0,∞)^{k}$ is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Fečkan [Proc. Amer. Math. Soc. 136 (2008)]. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone $C ⊆ ℝ^{k}$. Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures?

Ograniczanie wyników

3 Studia Mathematica

3 Glöckner H.

1 Lucht L.

1 Neeb K.-H.

1 Porubský Š.

1 2012

1 2009

1 2002

Algebras whose groups of units are Lie groups

When unit groups of continuous inverse algebras are regular Lie groups

General Dirichlet series, arithmetic convolution equations and Laplace transforms