Helton-Howe Trace, Connes-Chern character and Quantization

TL;DR

This paper explores the Helton-Howe trace and Connes-Chern character for Toeplitz operators on weighted Bergman spaces, establishing local formulas and analyzing their behavior as the weight parameter grows large, using quantization techniques.

Contribution

It provides new local formulas for the Helton-Howe trace and Connes-Chern character, demonstrating their independence from the weight parameter and applying harmonic analysis and quantization methods.

Findings

Helton-Howe trace is weight-independent.

Derived local formulas for large weight limits.

Connected quantization with Toeplitz operator analysis.

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.