Kernel and torsion of the first stabilisation map

Determine the kernel of the first matrix-stabilisation homomorphism from the index group of the operator algebra on the Banach space E=X\oplus X to the index group of its 2-by-2 matrix algebra, and establish whether the class [I_E-2ST] is torsion, where X and S,T are the Motakis realisation and operators supplied by Theorem A.

Background

The paper associates failures of exponential-spectral commutativity with elements in the kernel of the first matrix-stabilisation map on the index group Inv(A)/Inv_0(A). For the specific operators S,T constructed in Theorem A on E=X\oplus X, the class [I_E-2ST] lies in this kernel. Its image in the corresponding Calkin index group has order two, but the paper does not determine the structure of the full kernel or whether the class itself has finite order.