This is an announcement for the paper "Approximation properties of locally convex spaces and the problem of uniqueness of the trace of a linear operator" by Grigory L. Litvinov.
Abstract: In the present article, it is proved that every nuclear operator in a locally convex space E has a well-defined trace if E possesses the approximation property. However, even if a space possesses the approximation property this still does not guarantee a positive solution of A. Grothendieck's uniqueness problem for this space. Below, we present an example of a quasi-complete space with the approximation property in which it is not possible to define the trace for all Fredholm operators (in the sense of A. Grothendieck). We prove that the uniqueness problem has a positive solution if E possesses the "bounded approximation property." Preliminary information and results are presented in Section 2. A number of approximation-type properties of locally convex spaces and relations between these properties are considered in Section 3. The principal results of the present study, along with certain corollaries from these results (for example, the existence of a matrix trace), may be found in Section 4.
Archive classification: math.FA math.OA
Mathematics Subject Classification: 46A32, 46A35
Citation: Selecta Mathematica Sovietica, vol. 11, No.1 (1992), p. 25-40
Remarks: 18 pages
Submitted from: glitvinov@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1108.1721
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