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Search: MSC category 46M40 ( Inductive and projective limits [See also 46A13] )

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1. CJM Online first

Glöckner, Helge
Completeness of infinite-dimensional Lie groups in their left uniformity
We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$ such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on~$G_n$, for each $n\in\mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each~$G_n$ whenever $\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood in~$G$ for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover, each $G_n$ is complete, then~$G$ is complete. We also show that the weak direct product $\bigoplus_{j\in J}G_j$ is complete for each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$. As a consequence, every strict direct limit $G=\bigcup_{n\in \mathbb{N}}G_n$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\operatorname{Diff}_c(M)$ of a paracompact finite-dimensional smooth manifold~$M$ and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$ and complete Lie group~$H$ modelled on a complete locally convex space.

Keywords:infinite-dimensional Lie group, left uniform structure, completeness
Categories:22E65, 22A05, 22E67, 46A13, 46M40, 58D05

2. CJM 2004 (vol 56 pp. 3)

Amini, Massoud
Locally Compact Pro-$C^*$-Algebras
Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. Of these, the second and third are $C^*$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-$C^\ast$-algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the $C^\ast$-algebra $C_0(X)$, one can get the other three algebras by $C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl( K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of $C_0(X)$, respectively. In this article we consider the possibility of these transitions for general $C^\ast$-algebras. The difficult part is to start with a pro-$C^\ast$-algebra $A$ and to construct a $C^\ast$-algebra $A_0$ such that $A=\Gamma\bigl(K(A_0)\bigr)$. The pro-$C^\ast$-algebras for which this is possible are called {\it locally compact\/} and we have characterized them using a concept similar to that of an approximate identity.

Keywords:pro-$C^\ast$-algebras, projective limit, multipliers of Pedersen's ideal
Categories:46L05, 46M40

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