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Commutators in ideals of trace class operators. (English) Zbl 0602.47032
For any $p>0$, let $C\sb p$ 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 $K\sb 1(C\sb p,B)$ be $GL\sb 1(C\sb p)/[GL\sb 1(C\sb p),GL\sb 1(B)]$ (where $K\sb 1$ is a functor of algebraic K-theory) and $H\sp 1(C\sb p,B)$ be the additive group of $C\sb p$ modulo the subgroup generated by all additive commutators XJ-JX with X in $C\sb p$ and J in B (here $H\sp 1$ is a cyclic cohomology functor). When $p>1$, it has been known that both groups are trivial. When $p\le 1$, the group $K\sb 1(C\sb p,B)$ maps onto the multiplicative group $GL\sb 1{\bbfC}$ (determinant) and the group $H\sp 1(C\sb p,B)$ maps onto the additive group of the complex numbers ${\bbfC}$ (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 ${\bbfC}$ of uncountable dimension. The case $p\le 1$ was finished in part II [review below]: in this case the kernels are trivial.

##### MSC:
 47L30 Abstract operator algebras on Hilbert spaces 47B10 Operators belonging to operator ideals 47B47 Commutators, derivations, elementary operators, etc. 46M20 Methods of algebraic topology in functional analysis 46L80 $K$-theory and operator algebras
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