The Next Colloquium
Thursday, 26th of April 2012, 4pm, Theatre C
Robert Archbold
(University of Aberdeen)
Measuring non-commutativity in operator algebras
Let \(A\) be a non-commutative normed algebra with centre \(Z(A)\). A simple application of the triangle inequality shows that if \(a\in \)A is close to \(Z(A)\) then \(a\) almost commutes with elements of the unit ball of \(A\) and so the inner derivation implemented by \(a\) has small norm. The extent to which the converse holds is a basic question in the study of non-commutativity and has been investigated extensively in the case of von Neumann algebras and other \(C^*\)-algebras. In this context, \(K(A)\) is defined to be the smallest number in \([0,\infty]\) such that \[ {\rm distance}(a,Z(A)) \leq K(A)\Vert{\rm ad}(a)\Vert \] for all \(a\in A\), where \({\rm ad}(a)\) is the inner derivation \(x\to xa-ax\) \((x\in A)\).
We shall describe some old and new results in this area and give some indication of their dependence on
(i) the constrained optimization of the bounding radius of compact subsets of the plane,
(ii) primal ideals and the quantification of the failure of the Hausdorff property in the primitive ideal space \({\rm Prim}(A)\),
(iii) some spectral theory.
Forthcoming Colloquia - 2011 to 2012
Thursday, 10th of May 2012, 4pm, Theatre C
Alagacone Sri Ranga
(Universidade Esadual Paulista)
to be announced