Rieffel projections and 2-by-2 matrices

Abstract For a compact space Y , we view C ( Y × S 1 ) as the crossed product C ( Y ) ⋊ ℤ, with ℤ acting trivially. This allows us to study Rieffel projections in M 2 ( C ( Y × S 1 ): we characterize them and compute their image under the projection ∂ 0 : K 0 ( C ( Y × S 1 )) → K 1 ( C ( Y )). We provide a new Rieffel projection in M 2 ( C (𝕋 2 )): in contrast with Loring’s projection [16], which involves nonalgebraic functions, ours involves only trigonometric polynomials plus the square root of 2 − e 2 π i θ − e − 2 π i θ .We give applications of this projection, for example, explicit generators for the K-theory of C (𝕋 3 ). Finally, we prove that if a Banach algebra completion ℬ of C[ℤ n ] is continuously contained in C (𝕋 n ) and such that the Fourier series of (2 − e 2 π i θj − e − 2 π i θj )1 / 2 ( j = 1 , . . . , n ) converges in ℬ, then the inclusion ℬ ↪ C (𝕋 n ) induces isomorphisms in K-theory.