*-envelopping algebras from stochastic automata of topological subshifts: integrable dynamical system of newly weakened hyperbolocity

*-envelopping algebras of the functionals of stochastic automata from topological subshifts are newly written as the restriction of all the functions of isometric isomorphism on the weak open set. The Margulis Theory is here applied to theweak open set: the sigma-algebra generated from the weak topology coincides when transforming a strong topology: the original functional is smooth (therefore, a homeomorphism in the weak topology is a diffeomorphism in strong topology). The Frechet operator is descended to ordinary differentials in finite-dimensional spaces, after which the measure from the functional is written based on the Radon-Nikodym derivative dμ/dv: the transition probabilities are the derivatives of the functionals, which are density functions with respect to the measure v. The Frechet operator is a map that is used to define isomorphisms between Frechet spaces; after having constructed the isomorphism using the weak *-topology in a dual space, this map is used to transfer the isomorphism to a weak-topology-norm space. The isometric isomorphism is a specification of the Rokhlin aperiodic endomorphism. The non-empty compact space, the distance to it, and the functional constitute a triplet, which forms a Smale space. The newly found dynamical system is an integrable dynamical system of newly 'weakened' hyperbolicity; this dynamical system is compared with that of the Ruelle diffeomorphism [D. Ruelle, Non-commutative algebras for hyperbolic diffeomorphisms, Invent. Math. 93, 1-13(1988).] when the Sinai–Ruelle–Bowen measure is substituted with the new opportune uniform-entropy measure. The new corresponding Miatello–Wallach family of meromorphic functions is written: the variances are indicated to be calculated after application of generic potential theory. The star-algebras are proven to be reduced to envelopping algebras, to which the Tomita–Takesaki paradigm is proven to apply.