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- W48811906 abstract "We study the local operator space structure of nuclear $C^*$-algebras. It is shown that a $C^*$-algebra is nuclear if and only if it is an $OL_{infty, la}$ space for some (and actually for every) $la > 6$. The $OL_infty$ constant $lambda$ provides an interesting invariant [ OL_infty (A) = inf{la: ~ A ~{rm is ~ an} ~ {OL}_{infty, la} ~ {rm space}} ] for nuclear $C^*$-algebras. Indeed, if $A$ is a nuclear $C^*$-algebra, then we have $1le OL_infty (A) le 6$, and if $A$ is a unital nuclear $C^*$-algebra with $OL_{infty} (A) le (frac {1+{sqrt 5}}2)^{frac 12}$, we show that $A$ must be stably finite. We also investigate the connection between the rigid $OL_{infty, 1^+}$ structure and the rigid complete order $OL_{infty, 1^+}$ structure on $C^*$-algebras, where the latter structure has been studied by Blackadar and Kirchberg in their characterization of strong NF $C^*$-algebras. Another main result of this paper is to show that these two local structrues are actually equivalent on unital nuclear $C^*$-algebras. We obtain this by showing that if a unital (nuclear) $C^*$-algebra is a rigid ${OL}_{infty, 1^+}$ space, then it is inner quasi-diagonal, and thus is a strong NF algebra. It is also shown that if a unital (nuclear) $C^*$-algebra is an ${OL}_{infty, 1^+}$ space, then it is quasi-diagonal, and thus is an NF algebra." @default.
- W48811906 created "2016-06-24" @default.
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- W48811906 date "2002-06-06" @default.
- W48811906 modified "2023-09-27" @default.
- W48811906 title "On ${OL}_{infty}$ structure of nuclear $C^*$-algebras" @default.
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