Camillo Trapani

Partial *-Algebras and Their Operator Realizations

Foreword. Introduction. I: Theory of Partial O*-Algebras. 1. Unbounded Linear Operators in Hilbert Spaces. 2. Partial O*-Algebras. 3.Commutative Partial O*-Algebras. 4. Topologies on Partial O*-Algebras. 5. Tomita Takesaki Theory in Partial O*-Algebras. II: Theory of Partial *-Algebras. 6. Partial *-Algebras. 7. *-Representations of Partial *-Algebras. 8. Well-behaved X>*-Representations. 9. Biweights on Partial *-Algebras. 10. Quasi *-Algebras of Operators in Rigged Hilbert Spaces. 11. Physical Applications. Outcome. Bibliography. Index.

QUASI *-ALGEBRAS OF OPERATORS AND THEIR APPLICATIONS

The main facts of the theory of quasi*-algebras of operators acting in a rigged Hilbert space are reviewed. The particular case where the rigged Hilbert space is generated by a self-adjoint operator in Hilbert space is examined in more details. A series of applications to quantum theories are discussed.

Some classes of topological quasi *-algebras

The completion A[τ ] of a locally convex ∗-algebra A[τ ] with not jointly continuous multiplication is a ∗-vector space with partial multiplication xy defined only for x or y ∈ A0, and it is called a topological quasi ∗-algebra. In this paper two classes of topological quasi ∗-algebras called strict CQ∗-algebras and HCQ∗-algebras are studied. Roughly speaking, a strict CQ∗-algebra (resp. HCQ∗-algebra) is a Banach (resp. Hilbert) quasi ∗-algebra containing a C∗-algebra endowed with another involution # and C∗-norm ‖ ‖#. HCQ∗-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a HCQ∗-algebra if and only if it contains a left Hilbert algebra with unit as a dense subspace. Further, we shall give a necessary and sufficient condition under which a strict CQ∗-algebra is embedded in a HCQ∗-algebra.

Partial Inner Product Spaces

Parabolic Evolution Equations and their Applications The semigroup methods are known as a powerful tool for analyzing nonlinear diffusion equations and systems. The author has studied abstract parabolic evolution equations and their applications to nonlinear diffusion equations and systems for more than 30 years. He gives first, after reviewing the theory of analytic semigroups, an overview of the theories of linear, semilinear and quasilinear abstract parabolic evolution equations as well as general strategies for constructing dynamical systems, attractors and stable-unstable manifolds associated with those nonlinear evolution equations. In the second half of the book, he shows how to apply the abstract results to various models in the real world focusing on various self-organization models: semiconductor model, activator-inhibitor model, B-Z reaction model, forest kinematic model, chemotaxis model, termite mound building model, phase transition model, and Lotka-Volterra competition model. The process and techniques are explained concretely in order to analyze nonlinear diffusion models by using the methods of abstract evolution equations. Features 7 Fills the gaps of existing literature 7 Presents rigorous mathematical theories 7 Applies results focusing on various self-organization models 7 Author has studied abstract parabolic evolution equations and their applications to nonlinear diffusion equations and systems for more than 30 years Fields of interest Partial Differential Equations; Dynamical Systems and Ergodic Theory; Mathematical Biology in General Target groups Graduate students and researchers in mathematics, mathematical biology, mathematical ecology, and science of self-organization Type of publication Monograph

The Heisenberg dynamics of spin systems: A quasi*‐algebras approach

The problem of the existence of the thermodynamical limit of the algebraic dynamics for a class of spin systems is considered in the framework of a generalized algebraic approach in terms of a special class of quasi*‐algebras, called CQ*‐algebras. Physical applications to (almost) mean‐field models and to bubble models are discussed.

Biweights on partial *-algebras

The notion of weight, familiar for C*- and W*-algebras, is extended, in two different ways, to partial *-algebras, and the corresponding Gel′fand-Naimark-Segal representation is constructed (leading in one case to operators on a partial inner product space). Physically relevant examples are presented in the case of a graded partial *-algebra.

Non-self-adjoint hamiltonians defined by Riesz bases

We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, we give conditions under which these Hamiltonians can be factorized in terms of generalized lowering and raising operators.

Derivations of quasi *-algebras

The spatiality of derivations of quasi *-algebras is investigated by means of representation theory. Moreover, in view of physical applications, the spatiality of the limit of a family of spatial derivations is considered.

V*‐algebras: A particular class of unbounded operator algebras

We consider a weak unbounded commutant for a set of unbounded operators and we examine op*‐algebras which coincide with some of their bicommutant. This class of op*‐algebras, called V*‐algebras, shows some properties close to those which hold true for bounded operator algebras.

Exponentiating derivations of quasi ∗-algebras: possible approaches and applications

The problem of exponentiating derivations of quasi ∗-algebras is considered in view of applying it to the determination of the time evolution of a physical system. The particular case where observables constitute a proper CQ∗-algebra is analyzed.