Inductive limits of interval algebras: Unitary orbits of positive elements
The purpose of this paper is to introduce a new invariant for \(C^*\)- algebras and show that for the class of \(C^*\)-algebras that can be build as inductive limits of sequences of finite direct sums of interval algebras [i.e. algebras of the form \(C[0,1 ]\otimes M_n (\mathbb{C})\)], the invariant is complete in the sense that two such \(C^*\)-algebras are isomorphic if and only if the corresponding invariants are. The invariant is constructed in a way very similar to the construction of \(K_0 (A)\), or rather to the underlying abelian semigroup whose Grothendieck group is \(K_0 (A)\). The essential difference is that projections are substituted by general positive elements of norm less than one. As ambly demonstrated in [\textit{K. Thomsen}, Am. J. Math. 116, No. 3, 605-620 (1994; Zbl 0814.46050)]\ the projections (or the ordered \(K_0\)-group) do not contain enough information to classify inductive limits of interval algebras, but with projections substituted by more general positive elements we can mimic the classification of AF-algebras; not only in the result, but also in the proofs. By allowing more complicated positive elements than projections we get more structure in the classifying invariant, structure which is trivial or non-existing for projections. Thus the invariant (for stable isomorphism) is not only an abelian semigroup, but also a metric space and a left module for the action of the canonical semigroup. While the module structure of the invariant is quite innocent and uncomplicated the other new feature, the metric, is more involved. Hence the actual calculation (in the sense of obtaining a reasonable insight in the structure) of the invariant requires development of new methods and results, some of which are described in [\textit{V. S. Sunder} and \textit{K. Thomsen}, Houston J. Math. 18, No. 1, 127-137 (1992; Zbl 0812.46057)]\ and in Section 3 below. It turns out that for AF-algebras, irrational rotation algebras and other \(C^*\)-algebras of real rank zero, the invariant introduced here is a metric completion of \(R_\mathbb{C} \otimes K_0 (A)_+\), where \(R_\mathbb{C}\) is the invariant for the \(C^*\)-algebra \(\mathbb{C}\). Thus for AF- algebras the invariant is a strightforward modification of \(K_0 (A)\) as it ought to be.
scientific article; zbMATH DE number 126337