Helton-Howe Trace, Connes-Chern character and Quantization

Abstract

We study the Helton-Howe trace and the Connes-Chern character for Toeplitz operators on weighted Bergman spaces via the idea of quantization. We prove a local formula for the large $t$-limit of the Connes-Chern character as the weight goes to infinity. And we show that the Helton-Howe trace of Toeplitz operators is independent of the weight $t$ and obtain a local formula for the Helton-Howe trace for all weighted Bergman spaces using harmonic analysis and quantization.

72 pages, revised version. Version 1 of this article is split into two independent articles, Version 2 of arXiv:2204.04337 and arXiv:2210.04148. The main results of Version 1 of this article are presented in Version 2. And the results about trace of semicommutators on the unit disk together with related topics in Version 1 are included and generalized in a new paper, arXiv:2210.04148.