|
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces. Some applications of Dixmier traces to noncommutative geometry are described in . ==Definition== If ''H'' is a Hilbert space, then ''L''1,∞(''H'') is the space of compact linear operators ''T'' on ''H'' such that the norm : is finite, where the numbers ''μ''''i''(''T'') are the eigenvalues of |''T''| arranged in decreasing order. Let :. The Dixmier trace Tr''ω''(''T'') of ''T'' is defined for positive operators ''T'' of ''L''1,∞(''H'') to be : where lim''ω'' is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties: *lim''ω''(''α''''n'') ≥ 0 if all ''α''''n'' ≥ 0 (positivity) *lim''ω''(''α''''n'') = lim(''α''''n'') whenever the ordinary limit exists *lim''ω''(''α''1, ''α''1, ''α''2, ''α''2, ''α''3, ...) = limω(''α''''n'') (scale invariance) There are many such extensions (such as a Banach limit of ''α''1, ''α''2, ''α''4, ''α''8,...) so there are many different Dixmier traces. As the Dixmier trace is linear, it extends by linearity to all operators of ''L''1,∞(''H''). If the Dixmier trace of an operator is independent of the choice of lim''ω'' then the operator is called measurable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dixmier trace」の詳細全文を読む スポンサード リンク
|