Abstract
It is shown how to endow a hierarchy of sets of binary patterns with the structure of an abstract, normed C*-algebra. In the course we also recover an intermediate connection with the words of a Dyck language and Tempereley-Lieb algebras for which we also find that an effective arithmetic code is possible albeit of greater complexity. We also discuss possible applications associated with signal theory and waveform engineering on possible ways to embed discrete computational structures in an analog continuum substrate.
Highlights
The notion of C*-algebras, with “C” originally standing for “closed”, originates in the work of Gelfand and Neimark [1], and later Segal [2] as an attempt to an abstract generalization of the work of Jordan and Von Neumann [3] on quantum algebras of operators associated with Heisenberg matrix mechanics
Together with the notion of finitary information structures as being fundamental in the understanding of quantum dynamics seem to imply a tighter connection between the two themes of quantum structures and general signal theory
One can notice the existence of some dispersed evidence in the literature for the existence of certain analog models of computation that appear to be of similar power with models of quantum computation
Summary
The notion of C*-algebras, with “C” originally standing for “closed”, originates in the work of Gelfand and Neimark [1], and later Segal [2] as an attempt to an abstract generalization of the work of Jordan and Von Neumann [3] on quantum algebras of operators associated with Heisenberg matrix mechanics. We ask to construct the equivalent of some discrete normed division algebra over binary words of constant length in any Wk via actions on their integer values associated via the polynomial representation over a complex field C This calls for the separate definition of an addition and multiplication for word patterns plus the existence of an involutive conjugation operation, usually denoted as (*) and an appropriate norm ... To remedy the degenerate structure of the previous choice, we need to find another method of characterization of all patterns by a sufficiently sensitive measure of each pattern’s complexity and structure As is evident, such cannot be given in terms of the binary Shannon’s entropy since for any ( ) finite string, the particular function depends solely on a probability defined as. We examine some interesting relations between the conjugation operation and the alternative theme of the Tempereley-Lieb algebras
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