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

Balanced pairs of operators and their relative index