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    •   RIUMA Principal
    • Investigación
    • Álgebra, Geometría y Topología - (AGT)
    • AGT - Conferencias Científicas
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    •   RIUMA Principal
    • Investigación
    • Álgebra, Geometría y Topología - (AGT)
    • AGT - Conferencias Científicas
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    Primitive graph algebras

    • Autor
      Abrams, Gene
    • Fecha
      2014-06-18
    • Resumen
      Let $E = (E^0, E^1, s,r)$ be an arbitrary directed graph (i.e., no restriction is placed on the cardinality of $E^0$, or of $E^1$, or of $s^{-1}(v)$ for $v\in E^0$). Let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in a field $K$, and let $C^*(E)$ denote the graph C$^*$-algebra of $E$. % (Note: here $C^*(E)$ need not be separable.) We give necessary and sufficient conditions on $E$ so that $L_K(E)$ is primitive. (This is joint work with Jason Bell and K.M. Rangaswamy.) We then show that these same conditions are precisely the necessary and sufficient conditions on $E$ so that $C^*(E)$ is primitive. (This is joint work with Mark Tomforde.) This situation gives yet another example of algebraic / analytic properties of the graph algebras $L_K(E)$ and $C^*(E)$ for which the graph conditions equivalent to said property are identical, but for which the proof / techniques used are significantly different. In the Leavitt path algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive von Neumann regular algebras (thereby giving a systematic answer to a decades-old question of Kaplansky). In the graph C$^*$-algebra setting, we show how this result allows for the easy construction of a large collection of prime, non-primitive C$^*$-algebras (thereby giving a systematic answer to a decades-old question of Dixmier).
    • URI
      http://hdl.handle.net/10630/7688
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    CartelConfAbrams2014.pdf (127.9Kb)
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    • AGT - Conferencias Científicas

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