Partitions of unity and coverings

Austin, Kyle; Dydak, Jerzy

doi:10.1016/j.topol.2014.05.015

Cited by 2 publications

(1 citation statement)

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“…Given an open s-scale U, one can create a decreasing sequence of open s-scales U n , n ≥ 1, so that U 1 = U. There is a simple proof in [2] (see Theorem 3.4) that in such a case there is a partition of unity subordinated to U, i.e. a family of non-negative continuous functions {φ U } U∈U adding up to 1 with the support of each φ U contained in U .…”

Section: Scale Structuresmentioning

confidence: 99%

Scale Structures and C*-algebras

Austin¹,

Dydak²,

Holloway³

2016

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The purpose of this paper is to investigate the duality between large scale and small scale. It is done by creating a connection between C*-algebras and scale structures. In the commutative case we consider C*-subalgebras of C b (X), the C*-algebra of bounded complexvalued functions on X. Namely, each C*-subalgebra C of C b (X) induces both a small scale structure on X and a large scale structure on X. The small scale structure induced on X corresponds (or is analogous) to the restriction of C b (h(X)) to X, where h(X) is the Higson compactification. The large scale structure induced on X is a generalization of the C 0 -coarse structure of N.Wright. Conversely, each small scale structure on X induces a C*-subalgebra of C b (X) and each large scale structure on X induces a C*-subalgebra of C b (X). To accomplish the full correspondence between scale structures on X and C*-subalgebras of C b (X) we need to enhance the scale structures to what we call hybrid structures. In the noncommutative case we consider C*-subalgebras of bounded operators B(l 2 (X)).

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Section: Scale Structuresmentioning

confidence: 99%