Jan Spakula (Southampton)
Quasi-locality and Property A
Let X be a countable discrete metric space, and think of operators on L^2(X) in terms of their X-by-X matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of X. This algebra "encodes" the large-scale (a.k.a. coarse) structure of X. Quasi-locality, coined by John Roe in '88, is a property of an operator on L^2(X), designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work. (Joint with A Tikuisis and J Zhang.)
In the talk, I will introduce basics of coarse geometry, Property A and Roe algebras. Then I will move on to quasi-locality and (hopefully) the main ingredients of our argument: If X has Property A, then any quasi-local operator actually belongs to the Roe algebra.