A complete invariant for \(AD\) algebras with real rank zero and bounded torsion in \(K_ 1\)
In the context of the theory of nuclear and separable \(C^*\)-algebras, the author deals in this paper with the class of \(AD\)-algebras -- inductive limits of finite direct sums of \(C^*\)-algebras of continuous matrix-valued functions over the circle, or over the interval, where in the interval case we require that the value over the end points is a scalar multiple of the unit matrix. The purpose is to show that when the \(AD\)-algebras in question have bounded torsion in \(K_1\), i.e., when only a finite number of integers occur as orders of torsion elements, there is a simple answer to the following problem suggested by the work of \textit{M. Dădărlat} and \textit{T. A. Loring} [Math. Ann. 305, No. 4, 601-616 (1996; Zbl 0857.46039)]: a combination of the \(\text{mod }K_0\)-groups and the classical \(K\)-groups forms an algebraic invariant for the class of \(AD\)-algebras. So, for each integer \(p>1\), such an algebraic invariant for \(C^*\)-algebras is considered. The invariant consists of \(K_0,K_1\) the \(K_0\)-group with \(Z/p\) coefficients, the order structures which these groups possess, and the natural maps between the three groups. The author proves that this invariant is complete for the class of \(AD\) algebras of real rank zero if \(p\) annihilates every torsion element of \(K_1\).
direct sums of \(C^*\)-algebras of continuous matrix-valued functions over the circle
scientific article; zbMATH DE number 927001