Abstract.
We study the K-theory of unital C *-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K *(A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K *(A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence.
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Hunton, J., Shchukin, M. The K-Theory of C *-Algebras with Finite Dimensional Irreducible Representations. Integr. equ. oper. theory 54, 89–96 (2006). https://doi.org/10.1007/s00020-004-1346-0
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DOI: https://doi.org/10.1007/s00020-004-1346-0