### Haar measure on linear groups over local skew fields.

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Let A be a locally convex, unital topological algebra whose group of units ${A}^{\times}$ is open and such that inversion $\iota :{A}^{\times}\to {A}^{\times}$ is continuous. Then inversion is analytic, and thus ${A}^{\times}$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then ${A}^{\times}$ 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 ${A}^{\times}$ is an analytic Lie group without...

We show by example that the associative law does not hold for tensor products in the category of general (not necessarily locally convex) topological vector spaces. The same pathology occurs for tensor products of Hausdorff abelian topological groups.

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

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we...

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form ${a}_{d}\ast {g}^{\ast d}+{a}_{d-1}\ast {g}^{\ast (d-1)}+\cdots +a\u2081\ast g+a\u2080=0$, where $a\u2080,...,{a}_{d}:\mathbb{N}\to \u2102$ 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 ${\sum}_{x\in X}f\left(x\right){e}^{-sx}$ ($s\in {\u2102}^{k}$), where $X\subseteq {[0,\infty )}^{k}$ is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...

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