Narrow Results

Let be a separable stable and prime C*-algebra. We show that every proper nontrivial ideal in (and hence every proper ideal in which properly contains ) is not stable.

Some consequences are counterexamples naturally occuring, in the non-σ-unital case, to Zhang's Dichotomy as well as the Hjelmborg–Rørdam theory of stability.

A smoothing operator for a unitary representation $\pi :G\to \mathrm{\text{U}}({\mathscr{H}})$ of a (possibly infinite dimensional) Lie group $G$ is a bounded operator $A:{\mathscr{H}}\to {\mathscr{H}}$ whose range is contained in the space ${{\mathscr{H}}}^{\infty }$ of smooth vectors of $(\pi ,{\mathscr{H}})$. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map ${\pi }^A:G\to B({\mathscr{H}}),g\mapsto \pi (g)A$ is smooth. For unitary representations $(\pi ,{\mathscr{H}})$ which are semibounded, i.e. there exists an element $x_0\in 𝔤$ such that all operators $id\pi (x)$ from the derived representation, for $x$ in a neighborhood of $x_0$, are uniformly bounded from above, we show that ${{\mathscr{H}}}^{\infty }$ coincides with the space of smooth vectors for the one-parameter group ${\pi }_{x_0}(t)=\pi (expt{x}_0)$. As the main application of our results on smoothing operators, we present a new approach to host $C^{\ast }$-algebras for infinite dimensional Lie groups, i.e. $C^{\ast }$-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of $G$. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.