Related papers: A note on a construction of J.F. Feinstein

We give a counterexample to a conjecture of S.E. Morris by showing that there is a compact plane set X such that R(X) has no non-zero, bounded point derivations but such that R(X) is not weakly amenable. We also give an example of a…

It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of Joel Feinstein. Our example is R(K) for a certain compact…

It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years ago. It is shown that the set K can be chosen such that, in…

We investigate the completeness and completions of the normed algebras $D^{(1)}(X)$ for perfect, compact plane sets $X$. In particular, we construct a radially self-absorbing, compact plane set $X$ such that the normed algebra $D^{(1)}(X)$…

In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of…

We give a survey of the known connections between regularity conditions and amenability conditions in the setting of uniform algebras. For a uniform algebra $A$ we consider the set, $A_{lc}$, of functions in $A$ which are locally constant…

In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the…

Let $X = G/\Gamma$ be a quotient of a real Lie group by a non-uniform lattice. Consider a one-parameter subgroup $F$ of $G$ that is $\operatorname{Ad}$-diagonalizable over $\mathbb{C}$ and whose action on $(X,m_X)$ is mixing. In this…

Let $G$ be a locally compact group. If $G$ is finite then the amenability constant of its Fourier algebra, denoted by ${\rm AM}({\rm A}(G))$, admits an explicit formula [Johnson, JLMS 1994]; if $G$ is infinite then no such formula for ${\rm…

We show that the $F$-jumping numbers of a pair $(X, \mathfrak a)$ in positive characteristic have no limit points whenever the symbolic Rees algebra of $-K_X$ is finitely generated outside an isolated collection of points. We also give a…

Let $X$ be a perfect, compact subset of the complex plane. We consider algebras of those functions on $X$ which satisfy a generalised notion of differentiability, which we call $\mathcal{F}$-differentiability. In particular, we investigate…

We construct a consistent example of a topological space $Y=X \cup \{\infty\}$ such that: 1) $Y$ is regular. 2) Every $G_\delta$ subset of $Y$ is open. 3) The point $\infty$ is not isolated, but it is not in the closure of any discrete…

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

We show that for every finite subgroup $G$ of $Aut(F_n)$, the fixed point subcomplex $X_n^G$ is contractible, where $F_n$ is the free group on $n$ letters and $X_n$ is the spine of ``auter space'' constructed by Hatcher and Vogtmann. In…

Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact,…

We let the central Fourier algebra, ZA(G), be the subalgebra of functions u in the Fourier algebra A(G) of a compact group, for which u(xyx^{-1})=u(y) for all x,y in G. We show that this algebra admits bounded point derivations whenever G…

It is shown that there exists a compact set $X$ in ${\bf C}^N$ ($N\geq 2$) such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$ has a dense set of invertible elements, a large Gleason part, and an abundance of…

In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational.…

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically…

Let $\frak{F}(X, A)$ be one of the Banach algebras $\hbox{Lip}(X, A)$ or $\hbox{lip}(X, A)$. In this paper, we show that $\frak{F}(X, A)$ is amenable if and only if $X$ is uniformly discrete and $A$ is amenable. We also prove that the…