Minimal Cuntz–Krieger Dilations and Representations of Cuntz–Krieger Algebras

Proceedings Mathematical Sciences

B V Rajarama Bhat¹  Joachim Zacharias³  Santanu Dey² 

[1] Indian Statistical Institute, R.V. College Post, Bangalore 0 0, India$$;Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, Friedrich-Ludwig-Jahn-Str. a, Greifswald, Germany$$;School of Mathematical Sciences, University of Nottingham, Nottingham, NG RD UK$$

关键词: Dilation;    commuting tuples;    complete positivity;    Cuntz algebras;    Cuntz–Krieger algebras.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Given a contractive tuple of Hilbert space operators satisfying certain 𝐴-relations we show that there exists a unique minimal dilation to generators of Cuntz–Krieger algebras or its extension by compact operators. This Cuntz–Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximal 𝐴-relation piece. We define a maximal piece more generally for a finite set of polynomials in 𝑛 noncommuting variables. We classify all representations of Cuntz–Krieger algebras $mathcal{O}_A$ obtained from dilations of commuting tuples satisfying 𝐴-relations. The universal properties of the minimal Cuntz–Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.

【 授权许可】

Unknown