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By a quasi‐representation of a group  G , we mean an approximately multiplicative map of  G  to the unitary group of a unital  C *‐algebra. A quasi‐representation induces a partially defined map at the level  K ‐theory.  In the early 1990s, Exel and Loring associated two invariants with almost‐commuting pairs of unitary matrices  u  and  v : one a  K ‐theoretic invariant, which may be regarded as the image of the Bott element in  K  0 ( C ( 2 )) under a map induced by quasi‐representation of ℤ 2  in  U ( n ); the other is the winding number in ℂ \ {0} of the closed path  t  ↦ det( tvu  + (1 −  t ) uv ). The so‐called Exel–Loring formula states that these two invariants coincide if ‖ uv  −  vu ‖ is sufficiently small.  A generalization of the Exel–Loring formula for quasi‐representations of a surface group taking values in  U ( n ) was given by the second‐named author. Here, we further extend this formula for quasi‐representations of a surface group taking values in the unitary group of a tracial unital  C *‐algebra.

Quasi‐representations of surface groups