PROJECTIONS, COMMUTATORS AND LIE IDEALS IN <i>C</i>^*-ALGEBRAS

On closed Lie ideals of certain tensor products of ‐algebras

“…Remark For any two unital ‐algebras A and B , it is known ([, Corollary 2]) that , where is the Haagerup tensor product (see ). By a result of Fack (see [, Theorem 2.16]), the Cuntz algebra is spanned by its commutators and, therefore, it has no tracial functionals. Further, since is cross norm (see ), is an isometric homomorphism, so the Banach algebra does not have any tracial functionals, as well.…”

“…Analysis of ideal structures of various tensor products of operator algebras has been an important project and a good deal of work has been done in this direction -see, for instance, [2,10,13,[15][16][17]25]. On the other hand, there also exists an extensive literature devoted towards the study of Lie ideals, directly as well as through ideals of the algebra, in pure as well as Banach and operator algebras -see [6,9,[18][19][20][21] and the references therein.…”

“…Analysis of ideal structures of various tensor products of operator algebras has been an important project and a good deal of work has been done in this direction -see, for instance, [2,10,13,[15][16][17]25]. On the other hand, there also exists an extensive literature devoted towards the study of Lie ideals, directly as well as through ideals of the algebra, in pure as well as Banach and operator algebras -see [6,9,[18][19][20][21] and the references therein.The analysis of closed Lie ideals in operator algebras is primarily motivated by the evident relationship between commutators, projections and closed Lie ideals in * -algebras. For instance, Pedersen ([22, Lemma 1]) showed that the closed subspace () and the * -subalgebra () generated by the set of projections  of a * -algebra are both closed Lie ideals of ; and, moreover, if is simple with a non-trivial projection and if has at most one tracial state then = (), i.e., the span of the projections is dense in ([22, Corollary 4]).…”

“…where ⊗ ℎ is the Haagerup tensor product (see [8]). By a result of Fack (see [19,Theorem 2.16]), the Cuntz algebra  2 is spanned by its commutators and, therefore, it has no tracial functionals. Further, since ‖ ⋅ ‖ ℎ is cross norm (see [8]),  2 ∋ → ⊗ 1 ∈  2 ⊗ ℎ  2 is an isometric homomorphism, so the Banach algebra  2 ⊗ ℎ  2 does not have any tracial functionals, as well.…”

Sums of compositions of pairs of projections

“…We mention here important papers by Stampfli [8] (who showed that every operator on infinite dimensional H is a sum of 8 idempotents), Fillmore [5] (who showed that every operator on infinite dimensional H is a sum of 64 square-zero operators and a linear combination of 257 orthogonal projections) and Pearcy and Topping [7] (who improved these results showing that every operator on infinite dimensional H is a sum of 5 idempotents, a sum of 5 square-zero operators and a linear combination of 16 orthogonal projections). For a deep survey on this subject see an expository paper by Marcoux [6].…”