Positive Weighted Koopman Semigroups on Banach lattice modules
In this research work, we introduce the notion of a positive weighted semigroup representation on a Banach lattice module over a group representation on a commutative Banach lattice algebra. One main theme of this work is the following: for topological dynamics, we obtain the abstract representation of the lattice of continuous sections vanishing at infinity of a topological Banach lattice bundle (over a locally compact space \(\Omega\)) as a structure which we call an AM \(m\)-lattice module over \(C_0(\Omega)\) on which every positive weighted semigroup representation over the Koopman group representation on \(C_0(\Omega)\) is isomorphic to a positive weighted Koopman semigroup representation induced by a unique positive semiflow on the underlying topological Banach lattice bundle (over the continuous flow on the base space \(\Omega\)). As a result, every positive dynamical Banach lattice bundle can be assigned uniquely to a certain positive dynamical m-lattice module and vice versa, which is the Gelfand-type theorem that we proved. In order to do this, we establish the two categories of (i) Banach lattice modules and their dynamics; and (ii) Banach lattice bundles and their dynamics. We pay special attention to the case of a topological positive \(\mathbb{R}_+\)-dynamical Banach lattice bundle by which we obtain the corresponding \(C_0\)-semigroup of positive weighted Koopman operators, and using the theory of strongly continuous semigroups of positive operators, we obtain results pertaining to properties of the generator, and spectral theory of this positive semigroup.