Narrow Results

In this current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra cannot be $L^1$-predual. We prove also that a nilpotent ideal in a Johnson pseudo-contractible Banach algebras with the approximation property is forced to be zero. Among other things, we characterize the property $F$ of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup $S$ such that $(E(S),\le )$ is uniformly locally finite, $l^1(S)$ has the property $F$ if and only if each maximal subgroup of $S$ is amenable, where $E(S)$ is the set of idempotents of $S$. Furthermore, we study the property $F$ of some $𝜃$-Lau product structures.

A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories ${𝒞}$ with the fusion rules of $\mathrm{\text{SO}}{(8)}_2$ we determine the finite group $G$ for which $\mathrm{\text{Rep}}(D^{\omega }G)$ is braided equivalent to ${𝒵}({𝒞})$. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point.