A Lie ideal in an associative algebra A over C is a linear subspace L for which the commutator [a,m]:=am-ma of a and m lies in L whenever a lies in A and m lies in L.
In this talk, we shall discuss joint work with Gerard Murphy describing the Lie ideals in certain well-studied classes of C*-algebras, including (amongst others) simple, unital AF C*-algebras, irrational rotation algebras, Bunce-Deddens algebras, and Cuntz algebras. The linear span [A,A] of the commutators in a C*-algebra A is clearly a Lie ideal. We shall describe more recent results expressing elements of [A,A] as a sum of a very small number of commutators, and show how to use such an expression to decompose an arbitrary element of A as a linear combination of a finite linear combination of orthogonal projections in A, with a bound on the number of projections depending only upon the algebra, and not upon the element.
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Last updated 15 April 2008. This page was created and is maintained by Stephen Wills (s.wills@ucc.ie).