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Category: Coarse Geometry

Metric spaces and C*-algebras II

In the previous post about metric spaces and C*-algebras we saw the definition of the uniform Roe algebra \(C^*_u(X)\) of a metric space \(X\) and discussed its remarkable property that two metric spaces (of bounded geometry) \(X\) and \(Y\) are coarsely equivalent if and only if their uniform Roe algebras \(C^*_u(X)\) and \(C^*_u(Y)\) are Morita … Continue reading "Metric spaces and C*-algebras II"
Author Alexander EngelPosted on 11.09.2023

Metric spaces and C*-algebras

Let \(X\) be a metric space of bounded geometry. The latter means that for all \(r > 0\) exists an \(n_r \in \mathbb{N}\) such that every ball in \(X\) of radius \(r\) has at most \(n_r\) elements. This especially implies that \(X\) is discrete. It might seem a bit strange for some to consider discrete … Continue reading "Metric spaces and C*-algebras"
Author Alexander EngelPosted on 24.05.2023

Uniform and coarse embeddings of Banach spaces

Let \(X\) and \(E\) be Banach spaces. The metrics on these spaces induce both uniform and coarse structures, and we can ask whether the following two statements are equivalent to each other: \(X\) uniformly embeds into \(E\). \(X\) coarsely embeds into \(E\). Let us first write down what it means for a map \(\phi \colon … Continue reading "Uniform and coarse embeddings of Banach spaces"
Author Alexander EngelPosted on 19.04.2023

Quasi-local operators

In the first post of this series we asked at the end two questions – in this post we start working towards the answers in the general setup of the third post of this series. Our setup from the third post is the following: We have a metric space \((X,d)\) and we consider a bounded, … Continue reading "Quasi-local operators"
Author Alexander EngelPosted on 27.07.2020

AMS Special Session on the Mathematics of John Roe

There was a Special Session on Coarse Geometry, Index Theory, and Operator Algebras: Around the Mathematics of  John Roe at the Spring Central and Western Joint Sectional Meeting of the AMS last weekend to which Christopher and I were invited to give talks. John Roe passed away last year. His personal webpage is still online … Continue reading "AMS Special Session on the Mathematics of John Roe"
Author Alexander EngelPosted on 27.03.2019

Operators of finite propagation

In the starting post of this series we considered infinite band matrices (with uniformly bounded entries) acting on infinite vectors and asked at the end the question how to determine whether a given matrix, which is not a band matrix, can be approximated by such. Today we provide the setup in order to answer this question properly in … Continue reading "Operators of finite propagation"
Author Alexander EngelPosted on 17.08.2018

Equivariant band matrices and Fourier series

Recall that in the first post of this series we claimed that there exists an infinite matrix \(T\) which is in the closure (in operator norm) of the band matrices with uniformly bounded entries, but for which we have \(\|T^{(R)}\| \to \infty\). Here \[T^{(R)}_{m,n} := \begin{cases} T_{m,n} & \text{ if } |m-n| \le R\\ 0 & … Continue reading "Equivariant band matrices and Fourier series"
Author Alexander EngelPosted on 11.07.2018

Norms of infinite matrices

This is the first post of a series of posts in which we will eventually venture deep into the realm of coarse geometry. But we will always be motivated by questions which are related to the one that we will discuss here. But our first steps into coarse geometry will be very gently: we will … Continue reading "Norms of infinite matrices"
Author Alexander EngelPosted on 14.05.2018

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This blog discusses all kind of things of potential interest to connoisseurs of Geometry at Infinity. If you want to publish some article, or know of anything interesting that you would like to see an article written about, just write to alexander.engel@uni-greifswald.de

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