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Minimal Representations of Cuntz Algebras

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Abstract

Suppose that On (2 ≤ n ≤ ∞) is a Cuntz algebra. There exists a number 1 ≤ k(π, Ω) ≤ ∞ such that if the cyclic representations (H, π, Ω) and (H′,π′, Ω′) of the Cuntz algebra On are unitary equivalent, k(π, Ω) = k(π′, Ω′). Applying the number, one can define a minimal representation of On and give a sufficient condition of the minimality for a representation of On. Moreover, the relations between minimal states and minimal representations of Cuntz algebras and the relations between the minimality and the irreducibility of the representation of On are investigated, respectively.

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Acknowledgements

The author would like to thank the anonymous referees for helpful comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P. R. China

    Ling Juan Ye & Li Ning Jiang

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  1. Ling Juan Ye
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  2. Li Ning Jiang
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Correspondence to Li Ning Jiang.

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The project is supported by National Natural Science Foundation of China (Grant Nos. 11871303 and 11901038)

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Ye, L.J., Jiang, L.N. Minimal Representations of Cuntz Algebras. Acta. Math. Sin.-English Ser. 36, 749–764 (2020). https://doi.org/10.1007/s10114-020-8543-x

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