In the language of K-theory, the supernatural number correspond to the K0 group of the algebra.
In a broader context of the classification program for simple separable nuclear C*-algebras, AT-algebras of real rank zero were shown to be completely classified by their K-theory, the Choquet simplex of tracial states, and the natural pairing between K0 and traces.
It is also known that, in general, crossed products arising from minimal homeomorphism on the Cantor set are simple AT-algebras of real rank zero.
For a supernatural number {nk}, the corresponding Bunce–Deddens algebra B({nk}) is the direct limit: One needs to define the embeddings These imbedding maps arise from the natural embeddings between C*-algebras generated by shifts with periodic weights.
Because the map from W(n) into W(nm) preserves the compact operators, it descends into an embedding β : Mn(C(T)) → Mnm(C(T)).
The image of the above operator T ∈ W(n) under the natural embedding is the following 2n × 2n operator matrix in W(2n): Therefore, the action of the βk on the generator is A computation with matrix units yields that and where So In this particular instance, βk is called a twice-around embedding.
The reason for the terminology is as follows: as z varies on the circle, the eigenvalues of Z2 traces out the two disjoint arcs connecting 1 and -1.
An explicit computation of eigenvectors shows that the circle of unitaries implementing the diagonalization of Z2 in fact connect the beginning and end points of each arc.
Because all finite-dimensional vector bundles over the circle are homotopically trivial, the K0 of Mr(C(T)), as an ordered abelian group, is the integers Z with canonical ordered unit r. According to the above calculation of the connecting maps, given a supernatural number {nk}, the K0 of the corresponding Bunce–Deddens algebra is precisely the corresponding dense subgroup of the rationals Q.
The (C*-)crossed product given by (A, G, σ), denoted by is defined to be the C*-algebra with the following universal property: for any covariant representation (π, U), the C*-algebra generated by its image is a quotient of The Bunce–Deddens algebras in fact are crossed products of the Cantor sets with a natural action by the integers Z.