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Smoothing operators and $C^*$-algebras for infinite dimensional Lie groups. (arXiv:1505.02659v2 [math.RT] UPDATED)
来源于:arXiv
A host algebra of a (possibly infinite dimensional) Lie group $G$ is a
$C^*$-algebra whose representations are in one-to-one correspondence with
certain continuous unitary representations $\pi \colon G \to \U(\cH)$. In this
paper we present a new approach to host algebras for infinite dimensional Lie
groups which is based on smoothing operators, i.e., operators whose range is
contained in the space $\cH^\infty$ of smooth vectors. Our first major result
is a characterization of smoothing operators $A$ that in particular implies
smoothness of the maps $\pi^A \colon G \to B(\cH), g \mapsto \pi(g)A$. The
concept of a smoothing operator is particularly powerful for representations
$(\pi,\cH)$ which are semibounded, i.e., there exists an element $x_0 \in\g$
for which all operators $i\dd\pi(x)$, $x \in \g$, from the derived
representation are uniformly bounded from above in some neighborhood of $x_0$.
Our second main result asserts that this implies that $\cH^\infty$ coincides
with the space 查看全文>>