We study certain principal actions on noncommutative C*-algebras. Our main examples are the - and -actions on the odd-dimensional quantum spheres, yielding as fixed-point algebras quantum lens spaces and quantum complex projective spaces, respectively. The key tool in our analysis is the relation of the ambient C*-algebras with the Cuntz-Krieger algebras of directed graphs. A general result about the principality of the gauge action on graph algebras is given.
We consider the stochastic differential equation
,
where , , are nonrandom continuous functions of t, X₀ is an initial random variable, is a Gaussian process and X₀, Y are independent. We give the form of the solution () to (0.1) and then basing on the results of Plucińska [Teor. Veroyatnost. i Primenen. 25 (1980)] we prove that () is a quasi-diffusion proces.
This mainly expository article is devoted to recent advances in the study of dynamical aspects of the Cuntz algebras 𝓞ₙ, n < ∞, via their automorphisms and, more generally, endomorphisms. A combinatorial description of permutative automorphisms of 𝓞ₙ in terms of labelled, rooted trees is presented. This in turn gives rise to an algebraic characterization of the restricted Weyl group of 𝓞ₙ. It is shown how this group is related to certain classical dynamical systems on the Cantor set. An identification...
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