One can sometimes find an approximate solution to a problem, and then wants to know if there is an honest solution nearby. For example, if an element a of a C^*-algebra is such that aa^*-1 and a^*a-1 are small, then there is a unitary u that is close to a.
An old problem (Halmos, 1968) asked the same thing for pairs of commuting unitaries: it u,v are unitaries, and if uv-vu is small, are there actually commuting unitaries near u and v?
The answer is no (Voicuelescu, 1983), but yes if a 'winding-number' obstruction vanishes (Gong-Lin 1998, Eilers-Loring-Pedersen 1999). I will explain what this has to do with K-theory and group representations, and how to generalize it.
Copyright. All Rights Reserved.