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Notes on the nuclearity of random subalgebras of O₂{\mathcal {O}_2}

Authors:	Bernhard Burgstaller

Institution:	(1) Institute of Mathematics, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany

Abstract:	There exists a separable exact C-algebra A which contains all separable exact C-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have ${\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^(x_1,\ldots,x_n) \mbox{ is nuclear})=0$ . Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra ${\mathcal {O}_2}$ such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have ${\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^(x_1,\ldots,x_n) \mbox{ is nuclear})=0$ . We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C(0, 1]^d) is d. B. Burgstaller was supported by the Austrian Schr?dinger stipend J2471-N12.

Keywords:	Random C*-algebra Nuclear Cuntz algebra
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