The soft torus and applications to almost commuting matrices (Q689738)

The ``soft torus'' \(A_ \varepsilon\) is defined to be the universal \(C^*\)-algebra generated by a pair of unitaries for which the commutator has norm less than or equal to \(\varepsilon\). We show that the \(K\)-theory of \(A_ \varepsilon\) is naturally isomorphic to the \(K\)-theory of the algebra of continuous functions on the two-torus, although these algebras are not homotopically equivalent. This result is applied to give a new proof of the equality of certain invariants associated to almost commuting unitary matrices, namely the \(K\)-theory invariant and the winding number invariant.