0000000000061707
AUTHOR
Atsushi Inoue
Partial O*-Algebras
This chapter is devoted to the investigation of partial O*-algebras of closable linear operators defined on a common dense domain in a Hilbert space. Section 2.1 introduces of O- and O*-families, O- and O*-vector spaces, partial O*-algebras and O*-algebras. Partial O*-algebras and strong partial O*-algebras are defined by the weak and the strong multiplication. Section 2.2 describes four canonical extensions (closure, full-closure, adjoint, biadjoint) of O*-families and defines the notions of closedness and full-closedness (self-adjointness, integrability) of O*-families in analogy with that of closed (self-adjoint) operators. Section 2.3 deals with two weak bounded commutants M′w and M′qw …
Commutative Partial O*-Algebras
This chapter is devoted to the integrability of commutative partial O*-algebras. Three notions of weak commutativity, commutativity and strong commutativity of an O*-vector space are defined and investigated. In Section 3.1, we analyze the relation between the integrability of weakly commutative O*-vector space M and the commutativity of the von Neumann algebra (M w ′ )′. In Section 3.2, we study the integrable extensions of partial O*-algebras. In Section 3.3, we describe another explicit example, namely, the partial O*-algebra M[S, T] generated by two weakly commuting symmetric operators S and T defined on a common dense domain in a Hilbert space. In particular, we investigate in detail t…
Induced and reduced unbounded operator algebras
The induction and reduction precesses of an O*-vector space \({{\mathfrak M}}\) obtained by means of a projection taken, respectively, in \({{\mathfrak M}}\) itself or in its weak bounded commutant \({{\mathfrak M}^\prime_{\rm w}}\) are studied. In the case where \({{\mathfrak M}}\) is a partial GW*-algebra, sufficient conditions are given for the induced and the reduced spaces to be partial GW*-algebras again.
Unbounded Linear Operators in Hilbert Spaces
In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and…
Completely positive invariant conjugate-bilinear maps on partial *-algebras
The notion of completely positive invariant conjugate-bilinear map in a partial *-algebra is introduced and a generalized Stinespring theorem is proven. Applications to the existence of integrable extensions of *-representations of commutative, locally convex quasi*-algebras are also discussed.
Partial *-algebras of closable operators: A review
This paper reviews the theory of partial *-algebras of closable operators in Hilbert space (partial O*-algebras), with some emphasis on partial GW*-algebras. First we discuss the general properties and the various types of partial *-algebras and partial O*-algebras. Then we summarize the representation theory of partial *-algebras, including a generalized Gel’fand-Naimark-Segal construction; the main tool here is the notion of positive sesquilinear form, that we study in some detail (extendability, normality, order structure, …). Finally we turn to automorphisms and derivations of partial O*-algebras, and their mutual relationship. The central theme here is to find conditions that guarante…
Quasi *-Algebras of Operators in Rigged Hilbert Spaces
In this chapter, we will study families of operators acting on a rigged Hilbert space, with a particular interest in their partial algebraic structure. In Section 10.1 the notion of rigged Hilbert space D[t] ↪ H ↪ D × [t ×] is introduced and some examples are presented. In Section 10.2, we consider the space.L(D, D ×) of all continuous linear maps from D[t] into D × [t ×] and look for conditions under which (L(D, D ×), L +(D)) is a (topological) quasi *-algebra. Moreover the general problem of introducing in L(D, D ×) a partial multiplication is considered. In Section 10.3 representations of abstract quasi *-algebras into quasi*-algebras of operators are studied and the GNS-construction is …
Locally convex quasi $C^*$-normed algebras
Abstract If A 0 [ ‖ ⋅ ‖ 0 ] is a C ∗ -normed algebra and τ a locally convex topology on A 0 making its multiplication separately continuous, then A 0 ˜ [ τ ] (completion of A 0 [ τ ] ) is a locally convex quasi ∗-algebra over A 0 , but it is not necessarily a locally convex quasi ∗-algebra over the C ∗ -algebra A 0 ˜ [ ‖ ⋅ ‖ 0 ] (completion of A 0 [ ‖ ⋅ ‖ 0 ] ). In this article, stimulated by physical examples, we introduce the notion of a locally convex quasi C ∗ -normed algebra, aiming at the investigation of A 0 ˜ [ τ ] ; in particular, we study its structure, ∗-representation theory and functional calculus.
Unbounded C$^*$-seminorms and $*$-Representations of Partial *-Algebras
The main purpose of this paper is to construct *-representations from unbounded C*-seminorms on partial *-algebras and to investigate their *-representations. © Heldermann Verlag.
Representations and derivations of quasi ∗-algebras induced by local modifications of states
Abstract 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.
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 properCQ∗-algebra is analyzed.
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.
Weak commutation relations of unbounded operators: Nonlinear extensions
We continue our analysis of the consequences of the commutation relation $[S,T]=\Id$, where $S$ and $T$ are two closable unbounded operators. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. {We also consider what we call, adopting a physical terminology}, a {\em nonlinear} extension of the above commutation relations.
Well-behaved *-Representations
This chapter is devoted to the study of the so-called well-behaved *-representations of (partial) *-algebras. Actually one may define are two notions of well-behavedness and we will discuss the relation between them. These notions are introduced in order to avoid pathologies which may arise for general *-representations and to select “nice” representations, which may have a richer theory. In Section 8.1, we construct a class {π p } of *-representations, starting from an unbounded C*-seminorm p and we define nice *-representations in {π p }, called well-behaved. We also characterize their existence. In Section 8.2, we introduce the well-behaved *-representations associated with a compatible …
Bicommutants of reduced unbounded operator algebras
The unbounded bicommutant $(\mathfrak M_{E'})''$ of the {\em reduction} of an O*-algebra $\MM$ via a given projection $E'$ weakly commuting with $\mathfrak M$ is studied, with the aim of finding conditions under which the reduction of a GW*-algebra is a GW*-algebra itself. The obtained results are applied to the problem of the existence of conditional expectations on O*-algebras.
Non-self-adjoint resolutions of the identity and associated operators
Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $$\{X(\lambda )\}_{\lambda \in {\mathbb R}}$$ , whose adjoints constitute also a resolution of the identity, are studied. In particular, it is shown that a closed operator $$B$$ has a spectral representation analogous to the familiar one for self-adjoint operators if and only if $$B=\textit{TAT}^{-1}$$ where $$A$$ is self-adjoint and $$T$$ is a bounded inverse.
Tomita—Takesaki Theory in Partial O*-Algebras
This chapter is devoted to the development of the Tomita-Takesaki theory in partial O*-algebras. In Section 5.1, we introduce and investigate the notion of cyclic generalized vectors for a partial O*-algebra, generalizing that of cyclic vectors, and its commutants. Section 5.2 introduces the notion of a cyclic and separating system (M, λ, λ c ), which consists of a partial O*-algebra M, a cyclic generalized vector λ for M and the commutant λ c of λ. A cyclic and separating system (M, λ, λ c ) determines the cyclic and separating system ((M w ′ )′, λ cc , (λ cc ) c ) of the von Neumann algebra (M w ′ )′, and this makes it possible to develop the Tornita-Takesaki theory. Then λ can be extende…
Weak commutation relations of unbounded operators and applications
Four possible definitions of the commutation relation $[S,T]=\Id$ of two closable unbounded operators $S,T$ are compared. The {\em weak} sense of this commutator is given in terms of the inner product of the Hilbert space $\H$ where the operators act. Some consequences on the existence of eigenvectors of two number-like operators are derived and the partial O*-algebra generated by $S,T$ is studied. Some applications are also considered.
On the regularity of the partial {$O\sp *$}-algebras generated by a closed symmetric operator
Let be given a dense domain D in a Hilbert space and a closed symmetric operator T with domain containing D. Then the restriction of T to D generates (algebraically) two partial *-algebras of closable operators (called weak and strong), possibly nonabelian and nonassociative. We characterize them completely. In particular, we examine under what conditions they are regular, that is, consist of polynomials only, and standard. Simple differential operators provide concrete examples of all the pathologies allowed by the abstract theory.
*-Representations of Partial *-Algebras
This chapter is devoted to *-representations of partial *-algebras. We introduce in Section 7.1 the notions of closed, fully closed, self-adjoint and integrable *-representations. In Section 7.2, the intertwining spaces of two *-representations of a partial *-algebra are defined and investigated, and using them we define the induced extensions of a *-representation. Section 7.3 deals with vector representations for a *-representation of a partial *-algebra, which are the appropriate generalization to a *-representation of the notion of generalized vectors described in Chapter 5. Regular and singular vector representations are defined and characterized by the properties of the commutant, and…
Biweights on Partial *-Algebras
This chapter is devoted to the systematic investigation of biweights on partial *-algebras. These are a generalization of invariant positive sesquilinear forms that still allows a Gel’fand—Naĭmark—Segal (GNS) construction of representations. In Section 9.1, we apply this GNS construction for biweights and we obtain *-representations and cyclic vector representations of partial *-algebras, and we give some examples of biweights. Section 9.2 is devoted to the investigation of the Radon—Nikodým theorem and the Lebesgue decomposition theorem for biweights on partial *-algebras. In Section 9.3, we define regular and singular biweights on partial *-algebras and we characterize them with help of t…
Topologies on Partial O*-Algebras
In this chapter, we introduce some basic locally convex topologies on partial O*-algebras and we establish general properties of these topologies. In Section 4.1, we compare the graph topologies induced by different O-families on the same domain (and the corresponding families of bounded subsets). In the case where the domain D M of an O-family M is a (quasi-) Frechet space, the structure of bounded subsets in D M can be described in a rather explicit way. Section 4.2 and Section 4.3 are devoted to the topologization of (partial) O*-algebras. Section 4.2 deals with locally convex topologies, the so-called uniform topologies τ u , τ u , τ * u and quasiuniform topologies τ qu , and Section 4.…
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.
Structure of locally convex quasi C * -algebras
There are examples of C*-algebras A that accept a locally convex *-topology τ coarser than the given one, such that Ã[τ] (the completion of A with respect to τ) is a GB*-algebra. The multiplication of A[τ] may be or not be jointly continuous. In the second case, Ã[*] may fail being a locally convex *-algebra, but it is a partial *-algebra. In both cases the structure and the representation theory of Ã[τ] are investigated. If Ã+ τ denotes the τ-closure of the positive cone A+ of the given C*-algebra A, then the property Ā+ τ ∩ (-Ā+ τ) = {0} is decisive for the existence of certain faithful *-representations of the corresponding *-algebra Ã[τ]
Partial {$*$}-algebras of closable operators. II. States and representations of partial {$*$}-algebras
This second paper on partial Op*-algebras is devoted to the theory of representations. A new definition of invariant positive sesquilinear forms on partial *-algebras is proposed, which enables to perform the familiar GNS construction. In order to get a better control of the corresponding representations, we introduce and study a restricted class of partial Op*-algebras, called partial GW*-algebras, which turn up naturally in a number of problems. As an example, we extend Powers' results about the standardness of GNS representations of abelian partial *-algebras.
O* - Dynamical Systems and * - Derivations of Unbounded Operator Algebras
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.
Some classes of topological quasi *-algebras
The completion $\overline{A}[\tau]$ of a locally convex *-algebra $A [ \tau ]$ with not jointly continuous multiplication is a *-vector space with partial multiplication $xy$ defined only for $x$ or $y \in A_{0}$, 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 $\sharp$ and C$^*$-norm $\| \|_{\sharp}$. HCQ$^*$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a H…
Representable linear functionals on partial *-algebras
A GNS-like *-representation of a partial *-algebra \({{\mathfrak A}}\) defined by certain representable linear functionals on \({{\mathfrak A}}\) is constructed. The study of the interplay with the GNS construction associated with invariant positive sesquilinear forms (ips) leads to the notions of pre-core and of singular form. It is shown that a positive sesquilinear form with pre-core always decomposes into the sum of an ips form and a singular one.