Group C*-algebras satisfying Kadison's conjecture

Volume 96 / 2011

Rachid El Harti, Paulo R. Pinto Banach Center Publications 96 (2011), 147-157 MSC: Primary 46L05, 46L07; Secondary 43A07, 43A65. DOI: 10.4064/bc96-0-9

Abstract

We tackle R. V. Kadison's similarity problem (i.e. any bounded representation of any unital C$^*$-algebra is similar to a $^*$-representation), paying attention to the class of C$^*$-unitarisable groups (those groups $G$ for which the full group C$^*$-algebra C${^*}(G)$ satisfies Kadison's problem) and thereby we establish some stability results for Kadison's problem. Namely, we prove that $A\otimes_{\rm min} B$ inherits the similarity problem from those of the C$^*$-algebras $A$ and $B$, provided $B$ is also nuclear. Then we prove that $G/\Gamma$ is C$^*$-unitarisable provided $G$ is C$^*$-unitarisable and $\Gamma$ is a normal subgroup; and moreover, if $G/\Gamma$ is amenable and $\Gamma$ is C$^*$-unitarisable, so is the whole group $G$ ($\Gamma$ a normal subgroup).

Authors

  • Rachid El HartiDepartment of Mathematics and Computer Sciences
    Faculty of Sciences and Techniques, University Hassan I
    BP 577, 26000 Settat, Morocco
    e-mail
  • Paulo R. PintoCenter for Mathematical Analysis, Geometry, and Dynamical Systems
    Department of Mathematics, Instituto Superior Técnico
    Av. Rovisco Pais, 1049-001 Lisboa, Portugal
    e-mail

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