In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces X and Y , we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of X into the uniform Roe algebra of Y . Contents 1. Introduction 1 2. Preliminaries 5 3. Coarse-like maps and quasi-locality 8 4. Making embeddings strongly continuous 13 5. Embeddings and geometry preservation 16 6. Embeddings onto hereditary subalgebras 22 7. Digression on our geometric property 27 8. Open problems 30 References 32
Given a coarse space (X, E), one can define a C * -algebra C * u (X) called the uniform Roe algebra of (X, E). It has been proved by J. Špakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.
Abstract. In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotic uniform smoothness and convexity, and several distinct Banach-Saks-like properties. Among other results, we characterize the Banach spaces which are either coarsely or uniformly homeomorphic to T p 1 ⊕ . . . ⊕ T pn , where each T p j denotes the p j -convexification of the Tsirelson space, for p 1 , . . . , pn ∈ (1, . . . , ∞), and 2 ∈ {p 1 , . . . , pn}. We obtain applications to the coarse Lipschitz geometry of the p-convexifications of the Schlumprecht space, and some hereditarily indecomposable Banach spaces. We also obtain some new results on the linear theory of Banach spaces.
Abstract. Let SB be the standard coding for separable Banach spaces as subspaces of C(∆). In these notes, we show that if B ⊂ SB is a Borel subset of spaces with separable dual, then the assignment X → X * can be realized by a Borel function B → SB. Moreover, this assignment can be done in such a way that the functional evaluation is still well defined (Theorem 1). Also, we prove a Borel parametrized version of Zippin's theorem, i.e., we prove that there exists Z ∈ SB and a Borel function that assigns for each X ∈ B an isomorphic copy of X inside of Z (Theorem 5).
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