Contractive Representations and Toeplitz-Type Realizations of the Odometer Semigroup

Abstract: Let \(\mathcal O_n\) denote the odometer semigroup. We study its operator-theoretic representations, with particular emphasis on contractive, Nica-covariant, and Fock-space representations. We describe dilation and subrepresentation phenomena for \(\mathcal O_n\), and analyze the associated odometer maps \(W_L\) on vector-valued full Fock spaces. We obtain a canonical upper triangular decomposition in which the analytic component admits a Hardy-space realization as a Toeplitz operator \(M_\Theta\). This yields criteria for isometry, unitarity, invertibility, as well as norm identities and Douglas-type factorization properties. We also see characterizations of Fredholmness and essential normality, together with corresponding index and spectral consequences.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

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Noncommutative geometry in NYC

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