Balanced pairs of operators and their relative index

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.

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