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This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one-parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense, i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given. © 2019, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

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... Under some assumptions, the limit turns out to be a weak *-derivation generating a one parameter group of weak *-automorphisms, defined for a *-semisimple Banach quasi *-algebra, as we will investigate in the following section. For detailed discussion, see [4,6,20]. ...

... Then, for every fixed t ∈ R, β t (a) = e ith a e −ith is a well-defined weak *-automorphism of (A, A 0 ) since e ith , e −ith are bounded elements in A (see [4]) and thus (e ith a) e −ith = e ith (a e −ith ) for every a ∈ A. Moreover, β t is a uniformly bounded norm continuous group of weak *-automorphisms. The infinitesimal generator is given by ...

... For the tensor product Hilbert quasi *-algebra (H 1 ⊗ h H 2 , A 0 ⊗ B 0 ) we can apply all the known results for Banach quasi *-algebras about representability presented in Section 2, see also [4,5]. In particular, we know that ( ...

This survey aims to highlight some of the consequences that representable (and continuous) functionals have in the framework of Banach quasi *-algebras. In particular, we look at the link between the notions of *-semisimplicity and full representability in which representable functionals are involved. Then, we emphasize their essential role in studying *-derivations and representability properties for the tensor product of Hilbert quasi *-algebras, a special class of Banach quasi *-algebras.

We consider Sturm-Liouville problems with a boundary condition linearly dependent on the eigenparameter. We concentrate the study on the cases where non-real or non-simple (multiple) eigenvalues are possible. We prove that the system of root (i.e. eigen and associated) functions of the corresponding operator, with an arbitrary function removed, form a minimal system in L2(0, 1), except some cases where this system is neither complete nor minimal. The method used is based on the determination of the explicit form of the biorthogonal system. These minimality results can be extended to basis properties in L2(0, 1).

In the study of locally convex quasi *-algebras an important role is played by representable linear functionals; i.e., functionals which allow a GNS-construction. This paper is mainly devoted to the study of the continuity of representable functionals in Banach and Hilbert quasi *-algebras. Some other concepts related to representable functionals (full-representability, *-semisimplicity, etc) are revisited in these special cases. In particular, in the case of Hilbert quasi *-algebras, which are shown to be fully representable, the existence of a 1-1 correspondence between positive, bounded elements (defined in an appropriate way) and continuous representable functionals is proved.

Some structure properties of CQ * -algebras are investigated. The usual multiplication of a quasi * -algebra is generalized by introducing a weak- and strong-product. The * -semisimplicity is defined via a suitable family of positive sesquilinear forms and some consequences of this notion are derived. The basic elements of a functional calculus on these partial algebraic structures are discussed.

A normal Banach quasi *-algebra (script x sign, script u sign 0) has a distinguished Banach *-algebra script x sign b consisting of bounded elements of script x sign. The latter *-algebra is shown to coincide with the set of elements of script x sign having finite spectral radius. If the family P(script x sign) of bounded invariant positive sesquilinear forms on script x sign contains sufficiently many elements then the Banach *-algebra of bounded elements can be characterized via a C*-seminorm defined by the elements of P(script x sign).

The main aim of this paper is the investigation of conditions under which a locally convex quasi ⁎-algebra (A[τ],A0) attains sufficiently many (τ,tw)-continuous ⁎-representations in L†(D,H), to separate its points. Having achieved this, a usual notion of bounded elements on A[τ] rises. On the other hand, a natural order exists on (A[τ],A0) related to the topology τ, which also leads to a kind of bounded elements, which we call “order bounded”. What is important is that under certain conditions the latter notion of boundedness coincides with the usual one. Several nice properties of order bounded elements are extracted that enrich the structure of locally convex quasi ⁎-algebras.

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.

We review the main points in the development of partial *-algebras, at three dif-ferent levels: (i) The algebraic structure stemming from the partial multiplication; (ii) The locally convex partial *-algebras; (iii) The partial *-algebras of closable operators in Hilbert spaces or partial O*-algebras, including the representation theory of the abstract partial *-algebras.

We show a usefulness of the notion of dissipative operators in the study of derivations ofC*-algebras and prove that the closure of a normal *-derivation of UHF algebra satisfying a special condition is a generator of a one-parameter group of *-automorphisms.

We study unbounded derivations ofC*-algebras and characterize those which generate one-parameter groups of automorphisms. We also develop a functional calculus for the domains of closed derivations and develop criteria for closeability. Some specialC*-algebras are considered
\mathfrakB\mathbbC(\mathfrakH),\mathfrakB(\mathfrakH)\mathfrak{B}\mathbb{C}(\mathfrak{H}),\mathfrak{B}(\mathfrak{H})
, UHF algebras, and in this last context we prove the existence of non-closeable derivations.

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.

This book offers a review of the theory of locally convex quasi *-algebras, authored by two of its contributors over the last 25 years. Quasi *-algebras are partial algebraic structures that are motivated by certain applications in Mathematical Physics. They arise in a natural way by completing a *-algebra under a locally convex *-algebra topology, with respect to which the multiplication is separately continuous.
Among other things, the book presents an unbounded representation theory of quasi *-algebras, together with an analysis of normed quasi *-algebras, their spectral theory and a study of the structure of locally convex quasi *-algebras. Special attention is given to the case where the locally convex quasi *-algebra is obtained by completing a C*-algebra under a locally convex *-algebra topology, coarser than the C*-topology.
Introducing the subject to graduate students and researchers wishing to build on their knowledge of the usual theory of Banach and/or locally convex algebras, this approach is supported by basic results and a wide variety of examples.

Generalized B*-algebras are locally convex *-algebras which are generalizations of C*-algebras. We obtain results on unbounded derivations of commutative generalized B*-algebras (GB*-algebras for short) by borrowing some techniques from commutative algebra. We then give an example of a commutative GB*-algebra having a nonzero derivation. Lastly, we also prove that every derivation of a GB*-algebra, with underlying C*-algebra a W*-algebra, is inner.

It is an open question whether every derivation of a Fréchet GB
$^{\ast }$
-algebra
$A[{\it\tau}]$
is continuous. We give an affirmative answer for the case where
$A[{\it\tau}]$
is a smooth Fréchet nuclear GB
$^{\ast }$
-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB
$^{\ast }$
-algebras which are not pro-C
$^{\ast }$
-algebras.

The notion of ∗‐derivation on an algebra of unbounded operators is extended to partial O∗‐algebras, and the corresponding notion of spatiality is investigated. Special emphasis is given to ∗‐derivations associated to one‐parameter groups of ∗‐automorphisms and to ∗‐derivations of partial GW∗‐algebras.

The spatial theory of ∗‐automorphisms is well known for C∗‐ or W∗‐algebras and for algebras for unbounded operators on Hilbert spaces (O∗‐algebras). In this article, the theory is extended to partial ∗‐algebras of closable operators (partial O∗‐algebras), with many similar results.

Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.

This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made a considerable contribution. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of the C*-theory. The presentation, which is based on lectures given in Newcastle upon Tyne and Copenhagen, concentrates on topics involving quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, he globally constructs, within that setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment of the construction of time evolutions and KMS states.

The class of *-representations of a normed quasi *-algebra (X, U0) is investigated, mainly for its relationship with the structure of (X, U0). The starting point of this analysis is the construction of GNS-like *-representations of a quasi *-algebra (X, U 0) defined by invariant positive sesquilinear forms. The family of bounded invariant positive sesquilinear forms defines some seminorms (in some cases, C*-seminorms) that provide useful information on the structure of (X, U0) and on the continuity properties of its *- representations.

A spatial theory is developed for * - derivations of an algebra of unbounded operators, in terms of the concept of O*-dynamical systems. Three notions of spatiality emerge, depending on the nature of the corresponding generator. Special emphasis is put on O*-dynamical systems generated by one-parameter groups of *-automorphisms and their *-derivations.

It is well known that all derivations on a C*-algebra are continuous and that all derivations on a von Neumann algebra are
inner. The aim of this work is to provide some extensions of these results to the algebra of τ-measurable operators affiliated
with a von Neumann algebra equipped with a faithful semifinite normal trace τ.
Mathematics Subject Classification (2000)46H35-46L51
KeywordsAtomic projection lattice-inner derivation-derivation-τ-measurable operators-faithful semifinite normal trace-topology of convergence in measure-von Neumann algebra

It is investigated in which sense the Bogoliubov-Haag treatment of the B.C.S.-model gives the correct solution in the limit of infinite volume. We find that in a certain subspace of the infinite tensor product space the field operators show the correct time behaviour in the sense of strong convergence.

The relationship between the GNS representations associated to states on a quasi *-algebra, which are local modifications of each other (in a sense which we will discuss) is examined. The role of local modifications on the spatiality of the corresponding induced derivations describing the dynamics of a given quantum system with infinite degrees of freedom is discussed. I Introduction and preliminaries In two recent papers, [1, 2], we have investigated the role of derivations of quasi *-algebras and the possibility of finding a certain symmetric operator which implements the derivation, in the sense that in a suitable representation the derivation can be written as a commutator with an operator which in the physical literature is usually called the effective

We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, exponential stability and norm continuity of the solutions. Applications to a reaction-di usion equation with delay are given.

CQ * −algebras: structure properties

- F Bagarello
- C Trapani

Bagarello, F., Trapani, C.: CQ * −algebras: structure properties. Publ. RIMS, Kyoto Univ. 32, 85-116
(1996)

Università di Palermo, I-90123 Palermo, Italy E-mail address: mariastella.adamo@community.unipa.it; msadamo@unict.it Camillo Trapani, Dipartimento di Matematica e Informatica, Università di Palermo, I-90123 Palermo

- Maria Stella Adamo
- Dipartimento Di Matematica E Informatica

Maria Stella Adamo, Dipartimento di Matematica e Informatica,
Università di Palermo, I-90123 Palermo, Italy
E-mail address: mariastella.adamo@community.unipa.it; msadamo@unict.it
Camillo Trapani, Dipartimento di Matematica e Informatica, Università di Palermo, I-90123 Palermo, Italy
E-mail address: camillo.trapani@unipa.it

- A Pazy

Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied
Mathematical Sciences, vol. 44. Springer, New York (1983)