We characterize the class of RFD $C^{\ast }$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^{\ast }$-algebra is finite-dimensional, which is equivalent to the $C^{\ast }$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^{\ast }$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

In the spirit of the axiomatic approach by Rogers (1998) we show the equivalence between a set of assumptions on the behaviour of prices and the existence of a representation of these prices as conditional expectations. We rely on only weak assumptions and avoid any a priori modelling of negligible events or of any market filtration. Rather, both endogenously emerge along with the representation as conditional expectations.

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if $\unicode[STIX]{x1D6E4}$ is a nonamenable group and $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

Let $F$ be a non-Archimedean local field or a finite field. Let $n$ be a natural number and $k$ be 1 or 2. Consider $G\,:=\,\text{G}{{\text{L}}_{n+k}}\left( F \right)$ and let $M\,:=\,\text{G}{{\text{L}}_{n}}\left( F \right)\,\times \,\text{G}{{\text{L}}_{k}}\left( F \right)\,<\,G$ be a maximal Levi subgroup. Let $U\,<\,G$ be the corresponding unipotent subgroup and let $P\,=\,MU$ be the corresponding parabolic subgroup. Let $J\,:=\,J_{M}^{G}\,:\,\mathcal{M}\left( G \right)\,\to \,\mathcal{M}\left( M \right)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$. In this paper we prove that $J$ is a multiplicity free functor, i.e.,$\dim\,\text{Ho}{{\text{m}}_{M}}\left( J\left( \pi \right),\,\rho \right)\,\le \,1$, for any irreducible representations $\pi $ of $G$ and $\rho $ of $M$. We adapt the classical method of Gelfand and Kazhdan, which proves the “multiplicity free” property of certain representations to prove the “multiplicity free” property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

Starting with the extension of Lomonosov's Lemma by Tychonoff fixed point theorem, a result of Daughtry and Kim — Pearcy-Shields on rank-one commutators is extended to the context of locally convex spaces. Non-zero diagonal coefficients, eigenvalues and simultaneous triangular representations of compact operators on locally convex spaces are studied.