The kernel of the determinant map on certain simple C*-algebras

TL;DR

This paper characterizes the kernel of the de la Harpe--Skandalis determinant on certain simple C*-algebras, showing it consists precisely of elements that are products of eight multiplicative commutators.

Contribution

It provides a precise description of the kernel of the determinant on specific classes of simple C*-algebras, extending understanding of their algebraic structure.

Findings

Kernel of the determinant is exactly elements that are products of 8 multiplicative commutators

Results apply to unital separable simple C*-algebras with real rank zero, strict comparison, cancellation, or TAI property

Includes analogous results for the unitary case

Abstract

Let A be a unital separable simple C*-algebra such that either (1) A has real rank zero, strict comparison and cancellation or (2) A is TAI. We study the kernel of the de la Harpe--Skandalis determinant on GL^0(A), proving that the determinant vanishes exactly on elements which are products of 8 multiplicative commutators. We also have results for the unitary case.