Abstract
We study the noncommutative operator graph ℒ θ depending on a complex parameter θ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define a noncommutative group G and an algebra A θ which is the quotient of ℂG by a special algebraic relation depending on θ such that the matrix representation ϕ of A θ results in the algebra ℳ θ generated by ℒ θ . In the case of θ = ±1, the representation ϕ degenerates into a faithful representation of ℂK 4 , where K 4 is the Klein group. Thus, ℒ θ can be regarded as a noncommutative deformation of the graph associated with the Klein group. Bibliography: 16 titles.
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