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Subsets of the set of $g$-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, contained in the hull of a single point.
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In this article, we prove that a compact open set in the field $mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $mathbb{Z}/p^n mathbb{Z}$ ($pge 2$ prime, $nge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $mathbb{Q}_p$, some of which are self-similar measures.
The dual space of the C*-algebra of bounded uniformly continuous functions on a uniform space carries several natural topologies. One of these is the topology of uniform convergence on bounded uniformly equicontinuous sets, or the UEB topology for short. In the particular case of a topological group and its right uniformity, the UEB topology plays a significant role in the continuity of convolution. In this paper we derive a useful characterisation of bounded uniformly equicontinuous sets on locally compact groups. Then we demonstrate that for every locally compact group G the UEB topology on the space of finite Radon measures on G coincides with the right multiplier topology. In this sense the UEB topology is a generalisation to arbitrary topological groups of the multiplier topology for locally compact groups. In the final section we prove results about UEB continuity of convolution.
In this paper, we study product-free subsets of the free semigroup over a finite alphabet $A$. We prove that the maximum density of a product-free subset of the free semigroup over $A$, with respect to the natural measure that assigns a weight of $|A|^{-n}$ to each word of length $n$, is precisely $1/2$.
We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $Gamma$ as a spectral set, there is a one dimensional complex algebraic variety $Lambda$ in $Gamma$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $Lambda$. The variety $Lambda$ is shown to have a particular determinantal representation, related to the so-called fundamental operator of the pair (S,P). When (S,P) is a strict $Gamma$-contraction, then $Lambda$ is a distinguished variety in the symmetrized bidisc, i.e., a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary. We characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above.
Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exists a constant $lambda>0$ and an infinite sequence $(x_i)_{i=1}^inftysubset X$ such that $|x_i-x_j|=lambda$ for all $i eq j$.
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