Geodesic
Pairs of operators with equal defect from unitarity and their relative index
Sbornik. Mathematics, Tome 208 (2017) no. 10, pp. 1523-1534

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We show that the $K_1$ group of a $C^*$-algebra $A$ can be defined as the homotopy classes of pairs of matrices over $A$ that have equal defect from being unitary. We also consider pairs of pseudodifferential operators, not necessarily elliptic, with symbols forming a balanced pair. A relative index is defined for such pairs of operators, and it is proved to be equal to the topological index of the pair of symbols. Bibliography: 4 titles.
Keywords: $K$-theory, elliptic operator, index. 
V. M. Manuilov. Pairs of operators with equal defect from unitarity and their relative index. Sbornik. Mathematics, Tome 208 (2017) no. 10, pp. 1523-1534. http://geodesic.mathdoc.fr/item/SM_2017_208_10_a5/
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