### A characterisation of the circle group.

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We show that every compact connected group is the limit of a continuous inverse sequence, in the category of compact groups, where each successor bonding map is either an epimorphism with finite kernel or the projection from a product by a simple compact Lie group. As an application, we present a proof of an unpublished result of Charles Mills from 1978: every compact group is supercompact.

We study a method of approximating representations of the group $M\left(n\right)$ by those of the group $SO(n+1)$. As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of ${L}^{p}$ that applies to the “restrictions” of a function on the dual of $M\left(n\right)$ to the dual of $SO(n+1)$.

We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.The proof uses a criterion for absolute continuity involving ${L}^{p}$ spaces with $p\<1$: A measure $\mu $ on a compact metrisable group $K$ is absolutely continuous with respect to Haar measure $dk$ on $K$ if for some $p\<1$ a certain subspace of ${L}^{p}(K,dk)$ which is related to $\mu $ has sufficiently many continuous linear...