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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.

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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.

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.

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.

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 shown for the degenerate B.C.S.-model how in the limit of an infinite system the exact thermal Greens-functions approach a gauge invariant average of the one's calculated with the Bogoliubov-Haag method.

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.

- F Bagarello
- C Trapani

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- O Bratteli

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Derivations of τ -measurable Operators

- M Weigt

M. Weigt, Derivations of τ -measurable Operators, Operator Theory: Adv.
Appl., 195, (2009) 273286

Unbounded Derivations of GB*-algebras, Operator Theory Adv

- M Weigt
- I Zarakas

M. Weigt, I. Zarakas, Unbounded Derivations of GB*-algebras, Operator
Theory Adv. Appl. 247, (2015) 69-82.

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