Anderson, Joel; Vaserstein, L. N. Commutators in ideals of trace class operators. (English) Zbl 0602.47032 Indiana Univ. Math. J. 35, 345-372 (1986). For any \(p>0\), let \(C_ p\) be the Schatten p-class of operators. This is an ideal of the ring B of all bounded linear operators acting on a Hilbert space. Let \(K_ 1(C_ p,B)\) be \(GL_ 1(C_ p)/[GL_ 1(C_ p),GL_ 1(B)]\) (where \(K_ 1\) is a functor of algebraic K-theory) and \(H^ 1(C_ p,B)\) be the additive group of \(C_ p\) modulo the subgroup generated by all additive commutators XJ-JX with X in \(C_ p\) and J in B (here \(H^ 1\) is a cyclic cohomology functor). When \(p>1\), it has been known that both groups are trivial. When \(p\leq 1\), the group \(K_ 1(C_ p,B)\) maps onto the multiplicative group \(GL_ 1{\mathbb{C}}\) (determinant) and the group \(H^ 1(C_ p,B)\) maps onto the additive group of the complex numbers \({\mathbb{C}}\) (trace). It is proved in the paper that the kernels of these two maps are isomorphic (as groups). Moreover, when \(p=1\), these kernels are also isomorphic to the additive group of a vector space over \({\mathbb{C}}\) of uncountable dimension. The case \(p\leq 1\) was finished in part II [review below]: in this case the kernels are trivial. Cited in 1 ReviewCited in 8 Documents MSC: 47L30 Abstract operator algebras on Hilbert spaces 47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) 47B47 Commutators, derivations, elementary operators, etc. 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46L80 \(K\)-theory and operator algebras (including cyclic theory) Keywords:Schatten p-class of operators; algebraic K-theory; cyclic cohomology functor Citations:Zbl 0602.47033 PDFBibTeX XMLCite \textit{J. Anderson} and \textit{L. N. Vaserstein}, Indiana Univ. Math. J. 35, 345--372 (1986; Zbl 0602.47032) Full Text: DOI