کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6425401 | 1633803 | 2015 | 61 صفحه PDF | دانلود رایگان |

We introduce a new approach to traces on the principal ideal L1,â generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J. Dixmier, the new approach provides the explicit construction of every trace of every operator in L1,â in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on L1,â and the set of all translation invariant functionals on lâ. This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on lâ, and definitively classify the measurability of operators in L1,â in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class L1,â) from [7]. As an application we extend Connes' classical trace theorem to positive normalised traces.
Journal: Advances in Mathematics - Volume 285, 5 November 2015, Pages 568-628