Balanced pairs of operators and their relative index

TL;DR

This paper introduces a new way to define the K_1 group of a C*-algebra using balanced pairs of matrices and extends the concept to pairs of order zero pseudodifferential operators, establishing a relative index that matches the topological index.

Contribution

It provides a novel characterization of the K_1 group via balanced pairs and introduces a relative index for operator pairs that aligns with their symbolic topological index.

Findings

K_1 group characterized by homotopy classes of balanced pairs

Defined a relative index for pairs of order zero pseudodifferential operators

Established the equality of the relative index and the topological index

Abstract

We show that the K_1 group of a C*-algebra $A$ can be defined as homotopy classes of pairs, called balanced, of not necessarily unitary matrices over $A$ that have equal defects from being unitary. We also consider pairs of order zero pseudodifferential operators, not necessarily elliptic, with symbols being a balanced pair. A relative index is defined for such pairs of operators and it equals the topological index of the pair of their symbols.