Partial *-Algebras of Operators in a PIP-Space

The family of operators on a pip-space V is endowed with two, possibly different, partial multiplications, where partial means that the multiplication is not defined for any pair A,B of elements of Op(V) but only for certain couples. The two multiplications, to be called strong and weak, give rise to two different structures that coincide in certain situations. In this chapter we will discuss first the structure of Op(V) as partial *-algebra in the sense of [AIT02] and then the possibility of representing an abstract partial *-algebra into Op(V).

Operators on Partial Inner Product Spaces: Towards a Spectral Analysis

Given a LHS (Lattice of Hilbert spaces) $V_J$ and a symmetric operator $A$ in $V_J$, in the sense of partial inner product spaces, we define a generalized resolvent for $A$ and study the corresponding spectral properties. In particular, we examine, with help of the KLMN theorem, the question of generalized eigenvalues associated to points of the continuous (Hilbertian) spectrum. We give some examples, including so-called frame multipliers.

MR3063436 Reviewed Znojil, Miloslav; Wu, Junde A generalized family of discrete PT-symmetric square wells. Internat. J. Theoret. Phys. 52 (2013), no. 6, 2152–2162. (Reviewer: Camillo Trapani)

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 …

MR2888559 Muratov, M. A.; Chilin, V. I. (o)-topology in ∗-algebras of locally measurable operators. Ukrainian Math. J. 61 (2009), no. 11, 1798–1808. (Reviewer: Camillo Trapani)

MR3200323 Norling, Magnus Dahler, Inverse semigroup C∗-algebras associated with left cancellative semigroups. Proc. Edinb. Math. Soc. (2) 57 (2014), no. 2, 533–564.

Some seminorms on quasi*-algebras

Dieren t types of seminorms on a quasi -algebra (A; A0) are constructed from a suitable family F of sesquilinear forms on A. Two particular classes, extended C -seminorms and CQ -seminorms, are studied in some detail. A necessary and sucien t condition for the admissibility of a sesquilinear form in terms of extended C -seminorms on (A; A0) is given.

Operator (Quasi-)Similarity, Quasi-Hermitian Operators and All that

Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties. Then we study quasi-Hermitian operators, bounded or not, that is, operators that are quasi-similar to their adjoint and we discuss their application in pseudo-Hermitian quantum mechanics. Finally, we extend the analysis to operators in a partial inner product space (pip-space), in particular the scale of Hilbert space s generated by a single unbounded metric operator.

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…

Locally convex quasi *-algebras with sufficiently many *-representations

AbstractThe 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, that 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.

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…

Riesz-like bases in rigged Hilbert spaces

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{\xi_n\}\subset \D$ which is mapped by a one-to-one continuous operator $T:\D[t]\to\H[\|\cdot\|]$ into an orthonormal basis of the central Hilbert space $\H$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\H$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.

Biorthogonal vectors, sesquilinear forms, and some physical operators

Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized Riesz systems, enjoying certain properties. In particular we discuss what happens when they forms two $\D$-quasi bases.

Bounded and strongly bounded elements of Banach quasi *-algebras

Partial $\ast$-algebras of distributions

The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of C ∞ -vectors of a self-adjoint operator, is considered in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an

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.

Corrigendum: Partial inner product spaces, metric operators and generalized hermiticity

n/a

MR3050566 Calcagni, Gianluca; Nardelli, Giuseppe; Scalisi, Marco Quantum mechanics in fractional and other anomalous spacetimes. J. Math. Phys. 53 (2012), no. 10, 102110, 25 pp. (Reviewer: Camillo Trapani)

MR3126328 Dosi, Anar Quantum cones and their duality. Houston J. Math. 39 (2013), no. 3, 853–887

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…

MR2991872 Bender, Carl M.; Kuzhel, Sergii Unbounded C-symmetries and their nonuniqueness. J. Phys. A 45 (2012), no. 44, 444005, 14 pp. (Reviewer: Camillo Trapani)

Zbl 1286.46057 Dowerk, Philip A.; Savchuk, Yurii Induced *-representations and C∗-envelopes of some quantum *-algebras. J. Lie Theory 23, No. 1, 229-250 (2013).

MR3078826 Das, P. K.; Haldar, Prasanta Nonclassical states in interacting Fock space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013), no. 2, 1350015, 18 pp.

Closedness and lower semicontinuity of positive sesquilinear forms

The relationship between the notion of closedness, lower semicontinuity and completeness (of a quotient) of the domain of a positive sesquilinear form defined on a subspace of a topological vector space is investigated and sufficient conditions for their equivalence are given.

Representations of modules over a*-algebra and related seminorms

Representations of a module X over a � -algebra A# are considered and some related seminorms are constructed and studied, with the aim of finding bounded � -representations of A #.

Some Classes of Operators on Partial Inner Product Spaces

Many families of function spaces, such as $L^{p}$ spaces, Besov spaces, amalgam spaces or modulation spaces, exhibit the common feature of being indexed by one parameter (or more) which measures the behavior (regularity, decay properties) of particular functions. All these families of spaces are, or contain, scales or lattices of Banach spaces and constitute special cases of the so-called \emph{partial inner product spaces (\pip s)} that play a central role in analysis, in mathematical physics and in signal processing (e.g. wavelet or Gabor analysis). The basic idea for this structure is that such families should be taken as a whole and operators, bases, frames on them should be defined glo…

MR2522840 Kawamura, Katsunori Universal algebra of sectors. Internat. J. Algebra Comput. 19 (2009), no. 3, 347–371. (Reviewer: Camillo Trapani)

Metric operators, generalized hermiticity and partial inner product spaces

A quasi-Hermitian operator is an operator in a Hilbert space that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of metric operators, bounded or unbounded, in a Hilbert space. We introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties. Next we consider canonical lattices of Hilbert spaces generated by unbounded metric operators. Since such lattices constitute the simplest case of a partial inner product space (PIP space), we can exploit the te…

Lp-Spaces as Quasi *-Algebras

Abstract The Banach space L p ( X , μ), for X a compact Hausdorff measure space, is considered as a special kind of quasi *-algebra (called CQ*-algebra) over the C*-algebra C ( X ) of continuous functions on X . It is shown that, for p ≥2, ( L p ( X , μ),  C ( X )) is *-semisimple (in a generalized sense). Some consequences of this fact are derived.

Representations of modules over a *-algebra and related seminorms

Representations of a module X over a ∗-algebra A# are considered and some related seminorms are constructed and studied, with the aim of finding bounded ∗-representations of A#.

Partial {$*$}-algebras of closable operators. I. The basic theory and the abelian case

This paper, the first of two, is devoted to a systematic study of partial *-algebras of closable operators in a Hilbert space (partial Op*-algebras). After setting up the basic definitions, we describe canonical extensions of partial Op*-algebras by closure and introduce a new bounded commutant, called quasi-weak. We initiate a theory of abelian partial *-algebras. As an application, we analyze thoroughly the partial Op*-algebras generated by a single closed symmetric operator.

Unbounded derivations and *-automorphisms groups of Banach quasi *-algebras

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.

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 …

Bounded elements of C*-inductive locally convex spaces

The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: The first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to the natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.

The Partial Inner Product Space Method: A Quick Overview

Many families of function spaces play a central role in analysis, in particular, in signal processing (e.g., wavelet or Gabor analysis). Typical are spaces, Besov spaces, amalgam spaces, or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases ofpartial inner product spaces(PIP-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, and frames on them should be defined globally, for the whole family, instead o…

MR2728586 Dosi, Anar Local operator algebras, fractional positivity and the quantum moment problem. Trans. Amer. Math. Soc. 363 (2011), no. 2, 801–856. (Reviewer: Camillo Trapani)

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 biweights on partial *-algebras

Unbounded C*-seminorms generated by families of biweights on a partial *-algebra are considered and the admissibility of biweights is characterized in terms of unbounded C*-seminorms they generate. Furthermore, it is shown that, under suitable assumptions, when the family of biweights consists of all those ones which are relatively bounded with respect to a given C*-seminorm q, it can be obtained an expression for q analogous to that one which holds true for the norm of a C*-algebra.

Rigged Hilbert spaces and contractive families of Hilbert spaces

The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved that, when it exists, this rigged Hilbert space is the same as the canonical rigged Hilbert space associated to a family of closable operators in the central Hilbert space.

MR2851064 Daws, Matthew A bicommutant theorem for dual Banach algebras. Math. Proc. R. Ir. Acad. 111A (2011), no. 1, 21–28. (Reviewer: Camillo Trapani)

Quasi *-algebras and generalized inductive limits of C*-algebras

Banach partial *-algebras: an overview

A Banach partial $*$-algebra is a locally convex partial $*$-algebra whose total space is a Banach space. A Banach partial $*$-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of these objects and display a number of examples, namely, $L^p$-like function spaces and spaces of operators on Hilbert scales or lattices. Finally we analyze the important cases of Banach quasi $*$-algebras and $CQ^*$-algebras.

Singular Perturbations and Operators in Rigged Hilbert Spaces

A notion of regularity and singularity for a special class of operators acting in a rigged Hilbert space \({\mathcal{D} \subset \mathcal{H}\subset \mathcal{D}^\times}\) is proposed and it is shown that each operator decomposes into a sum of a regular and a singular part. This property is strictly related to the corresponding notion for sesquilinear forms. A particular attention is devoted to those operators that are neither regular nor singular, pointing out that a part of them can be seen as perturbation of a self-adjoint operator on \({\mathcal{H}}\). Some properties for such operators are derived and some examples are discussed.

Bounded elements in certain topological partial *-algebras

We continue our study of topological partial *algebras, focusing our attention to the interplay between the various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link between the strong and the weak multiplications, on one hand, and invariant positive sesquilinear (ips) forms, on the other. Then the analysis is extended to abstract topological partial *algebras, emphasizing the crucial role played by appropriate bounded elements, called $\M$-bounded. Finally, some remarks are made concerning representations in terms of the so-called partial GC*-algebras of operators.

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.

Esercizi di Analisi uno

MR2964711 Alfsen, Erik; Shultz, Fred Finding decompositions of a class of separable states. Linear Algebra Appl. 437 (2012), no. 10, 2613–2629. (Reviewer: Camillo Trapani)

Locally Convex Quasi C*-Algebras and Their Structure

Throughout this chapter \({{\mathfrak A}}_{\scriptscriptstyle 0}[\| \cdot \|{ }_{\scriptscriptstyle 0}]\) denotes a unital C*-algebra and τ a locally convex topology on \({{\mathfrak A}}_{\scriptscriptstyle 0}\). Let \(\widetilde {{{\mathfrak A}}_{\scriptscriptstyle 0}}[\tau ]\) denote the completion of \({{\mathfrak A}}_{\scriptscriptstyle 0}\) with respect to the topology τ. Under certain conditions on τ, a subspace \({\mathfrak A}\) of \(\widetilde {{{\mathfrak A}}_{\scriptscriptstyle 0}}[\tau ]\), containing \({{\mathfrak A}}_{\scriptscriptstyle 0}\), will form (together with \({{\mathfrak A}}_{\scriptscriptstyle 0}\)) a locally convex quasi *-algebra \(({\mathfrak A}[\tau ],{{\mathfrak…

Normed Quasi *-Algebras: Bounded Elements and Spectrum

Bounded elements of a Banach quasi *-algebra are intended to be those, whose images under every *-representation are bounded operators in a Hilbert space. This rough idea can be developed in several ways, as we shall see in the present chapter. These notions lead us to discuss a convenient concept of spectrum of an element in this context.

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. © 1996 American Institute of Physics.

Some results on the structure of partial*-algebras

Partial inner product spaces and operators on them

MR2777468 Volosova, Nina V. Contractible quantum Arens-Michael algebras. Banach algebras 2009, 423–440, Banach Center Publ., 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010. (Reviewer: Camillo Trapani)

MR3012631 Demirel, Semra The spectral shift function and Levinson's theorem for quantum star graphs. J. Math. Phys. 53 (2012), no. 8, 082110, 15 pp. (Reviewer: Camillo Trapani)

General Theory: Algebraic Point of View

It is convenient to divide our study of pip-spaces into two stages. In the first one, we consider only the algebraic aspects. That is, we explore the structure generated by a linear compatibility relation on a vector space V , as introduced in Section I.2, without any other ingredient. This will lead us to another equivalent formulation, in terms of particular coverings of V by families of subspaces. This first approach, purely algebraic, is the subject matter of the present chapter. Then, in a second stage, we introduce topologies on the so-called assaying subspaces \(\{V_r \}\). Indeed, as already mentioned in Section I.2, assuming the partial inner product to be nondegenerate implies tha…

Distributions Frames and bases

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal D$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fand-Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain $\mathcal D$ which acts like an orthonormal basis of the Hilbert space $\mathcal H$. The correspond…

Continuous *-homomorphisms of Banach Partial *-algebras

We continue the study of Banach partial *-algebras, in particular the question of the interplay between *-homomorphisms and biweights. Two special types of objects are introduced, namely, relatively bounded biweights and Banach partial *-algebras satisfying a certain Condition (S), which behave in a more regular way. We also present a systematic construction of Banach partial *-algebras of this type and exhibit several examples.

Locally Convex Quasi *-Algebras of Operators

This note is mainly concerned with locally convex quasi C*-normed *-algebras which arise as completions of C*-algebras of operators under certain topologies. Their importance is made clear by the representation theory of abstract locally convex quasi C*-normed *-algebras, investigated in previous papers and whose basic aspects are also overviewed here.

Partial Multiplication of Operators in Rigged Hilbert Spaces

The problem of the multiplication of operators acting in rigged Hilbert spaces is considered. This is done, as usual, by constructing certain intermediate spaces through which the product can be factorized. In the special case where the starting space is the set of C∞-vectors of a self-adjoint operator A, a general procedure for constructing a special family of interspaces is given. Their definition closely reminds that of the Bessel potential spaces, to which they reduce when the starting space is the Schwartz space \(\mathcal{S}(\mathbb{R}^n ).\) Some applications are considered.

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.

C*-seminorms on partial *-algebras: an overview

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.

*-Representations, seminorms and structure properties of normed quasi *-algebras

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.

MR2905225 Rowe, Stephen; Fang, Junsheng; Larson, David R. P1 subalgebras of Mn(C). Involve 4 (2011), no. 3, 213–250. (Reviewer: Camillo Trapani)

Reproducing pairs of measurable functions

We analyze the notion of reproducing pair of weakly measurable functions, which generalizes that of continuous frame. We show, in particular, that each reproducing pair generates two Hilbert spaces, conjugate dual to each other. Several examples, both discrete and continuous, are presented.

MR2784504 Winter, Wilhelm Strongly self-absorbing C∗-algebras are Z-stable. J. Noncommut. Geom. 5 (2011), no. 2, 253–264. (Reviewer: Camillo Trapani)

Weak A-frames and weak A-semi-frames

After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a weak lower $A$-semi-frame and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in (GB). We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.

Operators on PIP-Spaces and Indexed PIP-Spaces

As already mentioned, the basic idea of pip-spaces is that vectors should not be considered individually, but only in terms of the subspaces V r (r Є F), the building blocks of the structure. Correspondingly, an operator on a pipspace should be defined in terms of assaying subspaces only, with the proviso that only continuous or bounded operators are allowed. Thus an operator is a coherent collection of continuous operators. We recall that in a nondegenerate pip-space, every assaying subspace V r carries its Mackey topology \(\tau (V_r , V \bar{r})\) and thus its dual is \(V \bar{r}\). This applies in particular to \(V^{\#}\) and V itself. For simplicity, a continuous linear map between two…

Morphisms of certain banach C*-modules

Morphisms and representations of a class of Banach C*-modules, called CQ*algebras, are considered. Together with a general method for constructing CQ*-algebras, two different ways of extending the GNS-representation are presented.

Un modulo di Analisi due

MR2846469 Salmi, Pekka Quantum semigroup compactifications and uniform continuity on locally compact quantum groups. Illinois J. Math. 54 (2010), no. 2, 469–483. (Reviewer: Camillo Trapani)

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 …

MR2849476Proskurin, D. On C∗-algebra generated by Fock representation of Wick algebra with braided coefficients. Methods Funct. Anal. Topology 17 (2011), no. 2, 168–173. (Reviewer: Camillo Trapani)

Representations of Certain Banach C*-modules

The possibility of extending the well known Gelfand–Naimark– Segal representation of *-algebras to certain Banach C*-modules is studied. For this aim the notion of modular biweight on a Banach C*-module is introduced. For the particular class of strict pre CQ*-algebras, two different types of representations are investigated.

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.

MR2641584 Joiţa, Maria On extremal covariant completely multi-positive linear maps. Proceedings of the Sixth Congress of Romanian Mathematicians. Vol. 1, 337–345, Ed. Acad. Române, Bucharest, 2009. (Reviewer: Camillo Trapani)

Remarks on Infinite-Dimensional Representations of the Heisenberg Algebra

Infinite-dimensional representations of Lie algebras necessarily invoke the theory of unbounded operator algebras. Starting with the familiar example of the Heisenberg Lie algebra, we sketch the essential features of this interaction, distinguishing in particular the cases of integrable and nonintegrable representations. While integrable representations are well understood, nonintegrable representations are quite mysterious objects. We present here a short and didactical-minded overview of the subject.

Quasi *-Algebras and Multiplication of Distributions

AbstractA self-adjoint operatorAinL2(Ω,μ) defines in a natural way a space of test functions SA(Ω) and a corresponding space of distributions S′A(Ω). These are considered as quasi *-algebras and the problem of multiplying distributions is studied in terms of multiplication operators defined on a rigged Hilbert space.

Biweights and *-homomorphisms of partial *-algebras

Consider two partial *-algebras, 1 and 2, and an *-homomorphism Φ from 1 into 2. Given a biweight ϕ on 2, we discuss conditions under which the natural composition ϕ∘Φ of ϕ and Φ is a biweight on 1. In particular, we examine whether the restriction of a biweight to a partial *-subalgebra is again a biweight.

Normed Quasi *-Algebras: Basic Theory and Examples

In this chapter we shall consider the case, where \({\mathfrak A}\) is endowed with a norm topology, making \(({\mathfrak A},{{\mathfrak A}}_{\scriptscriptstyle 0})\) into a normed quasi *-algebra in the sense of Definition 3.1.1, below. This opens our discussion on locally convex quasi *-algebras, starting from the simplest situation. Nevertheless, as we shall see, simple does not mean trivial at all.

Erratum/Addendum to the paper ``Some seminorms on quasi*-algebras" (Studia Math. 158 (2003), 99–115)

Applications in Mathematical Physics

It turns out that pip-space methods have many applications in physics, although they are seldom mentioned as such. To draw on a literary analogy, like Moliere’s Monsieur Jourdain speaking in prose without knowing so, many authors have been using pip-space language without realizing it. In particular, chains or lattices of Hilbert spaces are quite common in many fields of mathematical physics. Some of these applications will be discussed at length in this chapter. To mention a few examples: quantum mechanics, in particular singular interactions (Section 7.1.3), scattering theory (Section 7.2), quantum field theory (Section 7.3), representations of Lie groups (Section 7.4), etc.

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.

Extensions of hermitian linear functionals

AbstractWe study, from a quite general point of view, the family of all extensions of a positive hermitian linear functional $$\omega $$ ω , defined on a dense *-subalgebra $${\mathfrak {A}}_0$$ A 0 of a topological *-algebra $${\mathfrak {A}}[\tau ]$$ A [ τ ] , with the aim of finding extensions that behave regularly. The sole constraint the extensions we are dealing with are required to satisfy is that their domain is a subspace of $$\overline{G(\omega )}$$ G ( ω ) ¯ , the closure of the graph of $$\omega $$ ω (these are the so-called slight extensions). The main results are two. The first is having characterized those elements of $${\mathfrak {A}}$$ A for which we can find a positive her…

CQ*-algebras: Structure properties

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 *-semisemplicity 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.

Erratumąddendum to the paper: ``Quasi*-algebras and generalized inductive limits of C*-algebras'' (Studia Math. 202 (2011), 165–190)

Algebraic dynamics in O*-algebras: a perturbative approach

In this paper the problem of recovering an algebraic dynamics in a perturbative approach is discussed. The mathematical environment in which the physical problem is considered is that of algebras of unbounded operators endowed with the quasiuniform topology. After some remarks on the domain of the perturbation, conditions are given for the dynamics to exist as the limit of a net of regularized linear maps. © 2002 American Institute of Physics.

PIP-Spaces and Signal Processing

Contemporary signal processing makes an extensive use of function spaces, always with the aim of getting a precise control on smoothness and decay properties of functions. In this chapter, we will discuss several classes of such function spaces that have found interesting applications, namely, mixed-norm spaces, amalgam spaces, modulation spaces, or Besov spaces. It turns out that all those spaces come in families indexed by one or more parameters, that specify, for instance, the local behavior or the asymptotic properties. In general, a single space, taken alone, does not have an intrinsic meaning, it is the family as a whole that does, which brings us to the very topic of this volume. In …

MR3183843 Bermúdez, Teresa; Martinón, Antonio; Negrín, Emilio On expectation of some products of Wick powers. J. Appl. Funct. Anal. 9 (2014), no. 1-2, 127–131.

States and representations of CQ∗ -algebras

A class of quasi *-algebras which exhibits some analogy with C*-algebras is studied. The extension of some properties of C*-algebras which are relevant for physical applications (such as the GNS-representation) is discussed. Quasi *-algebras of linear operators in rigged Hilbert space are shown to be typical examples of the developed framework.

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…

Examples of Indexed PIP-Spaces

This chapter is devoted to a detailed analysis of various concrete examples of pip-spaces. We will explore sequence spaces, spaces of measurable functions, and spaces of analytic functions. Some cases have already been presented in Chapters 1 and 2. We will of course not repeat these discussions, except very briefly. In addition, various functional spaces are of great interest in signal processing (amalgam spaces, modulation spaces, Besov spaces, coorbit spaces). These will be studied systematically in a separate chapter (Chapter 8).

A note on banach partial *-algebras

A Banach partial *-algebra is a locally convex partial *-algebra whose total space is a Banach space. A Banach partial *-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of such objects and display a number of examples, namely LP-like function spaces and spaces of operators on Hilbert scales.

Partial *-algebras of distributions

The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of analytic vectors of a self-adjoint operator is considered, in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an abstract version of the Bessel potential spaces, are used to factorize the product.

Faithfully representable topological *-algebras: some spectral properties

A faithfully representable topological *-algebra (fr*-algebra) A0 is characterized by the fact that it possesses sufficiently many *-representations. Some spectral properties are examined, by constructing a convenient quasi *-algebra A over A0, starting from the order bounded elements of A0.

Analisi Matematica - Funzioni di una variabile reale

General Theory: Topological Aspects

In Chapter 1, we have analyzed the structure of pip-spaces from the algebraic point of view only, (i.e., the compatibility relation). Here we will discuss primarily the topological structure given by the partial inner product itself. The aim is to tighten the definitions so as to eliminate as many pathologies as possible. The picture that emerges is reassuringly simple: Only two types of pip-spaces seem sufficiently regular to have any practical use, namely lattices of Hilbert spaces (LHS) or Banach spaces (LBS), that we have introduced briefly in the Introduction. Our standard reference on locally convex topological vector spaces (LCS) will be the textbook of Kothe [Kot69]. In addition, fo…

Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces

Pseudo-Hermitian quantum mechanics (QM) is a recent, unconventional, approach to QM, based on the use of non-self-adjoint Hamiltonians, whose self-adjointness can be restored by changing the ambient Hilbert space, via a so-called metric operator. The PT-symmetric Hamiltonians are usually pseudo-Hermitian operators, a term introduced a long time ago by Dieudonné for characterizing those bounded operators A that satisfy a relation of the form GA = A G, where G is a metric operator, that is, a strictly positive self-adjoint operator. This chapter explores further the structure of unbounded metric operators, in particular, their incidence on similarity. It examines the notion of similarity betw…

MR2666982 Heo, Jaeseong; Hong, Jang Pyo; Ji, Un Cig On KSGNS representations on Krein C∗-modules. J. Math. Phys. 51 (2010), no. 5, 053504, 13 pp. (Reviewer: Camillo Trapani)

SVEP and local spectral radius formula for unbounded operators

In this paper we study the localized single valued extension property for an unbounded operator T. Moreover, we provide sufficient conditions for which the formula of the local spectral radius holds for these operators.

Zbl 1282.39034 Park, Choonkil; Lee, Jung Rye; Shin, Dong Yun; Eshaghi Gordji, Madjid Comment on “Stability of (α,β,γ)-derivations on Lie C∗-algebras” by M. Eshaghi Gordji and N. Ghobadipour.

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.

MR2666967 Jäkel, Christian D.; Narnhofer, Heide; Wreszinski, Walter F. On the mixing property for a class of states of relativistic quantum fields. J. Math. Phys. 51 (2010), no. 5, 052703, 12 pp. (Reviewer: Camillo Trapani)

Gibbs states defined by biorthogonal sequences

Motivated by the growing interest on PT-quantum mechanics, in this paper we discuss some facts on generalized Gibbs states and on their related KMS-like conditions. To achieve this, we first consider some useful connections between similar (Hamiltonian) operators and we propose some extended version of the Heisenberg algebraic dynamics, deducing some of their properties, useful for our purposes.

A note on faithful traces on a von Neumann algebra

In this short note we give some techniques for constructing, starting from a {\it sufficient} family $\mc F$ of semifinite or finite traces on a von Neumann algebra $\M$, a new trace which is faithful.

Some remarks on quasi-Hermitian operators

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator.Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

Unbounded C*-seminorms, biweights and *-representations of partial *-algebras: a review

The notion of (unbounded) C*-seminorms plays a relevant role in the representation theory of *-algebras and partial *-algebras. A rather complete analysis of the case of *-algebras has given rise to a series of interesting concepts like that of semifinite C*-seminorm and spectral C*-seminorm that give information on the properties of *-representations of the given *-algebra A and also on the structure of the *-algebra itself, in particular when A is endowed with a locally convex topology. Some of these results extend to partial *-algebras too. The state of the art on this topic is reviewed in this paper, where the possibility of constructing unbounded C*-seminorms from certain families of p…

*-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…

Partial inner product spaces: Some categorical aspects

We make explicit in terms of categories a number of statements from the theory of partial inner product spaces (PIP spaces) and operators on them. In particular, we construct sheaves and cosheaves of operators on certain PIP spaces of practical interest.

Faithful representations of left C*-modules

The existence of a faithful modular representation of a left module $$ \mathfrak{X} $$ over a C*-algebra $$ \mathfrak{A}_\# $$ possessing sufficiently many traces is proved.

MR2544061 Ludkovsky, S. V. Algebras of operators in Banach spaces over the quaternion skew field and the octonion algebra. J. Math. Sci. (N. Y.) 144 (2007), no. 4, 4301–4366; translated from Sovrem. Mat. Prilozh. No. 35 (2005), 98--162(Russian) (Reviewer: Camillo Trapani)

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.…

Generalized Riesz systems and quasi bases in Hilbert space

The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ such that $\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.

Reproducing pairs of measurable functions and partial inner product spaces

We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product space (PIP spaces). Several examples, both discrete and continuous, are presented.

MR3010675 Emamirad, Hassan; Rogeon, Philippe Semiclassical limit of Husimi function. Discrete Contin. Dyn. Syst. Ser. S 6 (2013), no. 3, 669–676. (Reviewer: Camillo Trapani)

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.

MR2979407 Bougie, Jonathan; Gangopadhyaya, Asim; Mallow, Jeffry; Rasinariu, Constantin Supersymmetric quantum mechanics and solvable models. Symmetry 4 (2012), no. 3, 452–473. (Reviewer: Camillo Trapani)

C*-seminorms defined by families of positive sesquilinear forms

Representation Theorems for Solvable Sesquilinear Forms

New results are added to the paper [4] about q-closed and solvable sesquilinear forms. The structure of the Banach space $\mathcal{D}[||\cdot||_\Omega]$ defined on the domain $\mathcal{D}$ of a q-closed sesquilinear form $\Omega$ is unique up to isomorphism, and the adjoint of a sesquilinear form has the same property of q-closure or of solvability. The operator associated to a solvable sesquilinear form is the greatest which represents the form and it is self-adjoint if, and only if, the form is symmetric. We give more criteria of solvability for q-closed sesquilinear forms. Some of these criteria are related to the numerical range, and we analyse in particular the forms which are solvable…

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]

Order boundedness and spectrum in locally convex quasi *-algebras

After a quick sketch of the basic aspects of locally convex quasi *-algebras, we focus on order bounded elements and use them to analyze some spectral properties, trying to generalize the approach already studied in the Banach case.

Partial inner product spaces, metric operators and generalized hermiticity

Motivated by the recent developments of pseudo-hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism.

Refinements of PIP-Spaces

We have seen in Section 1.5, that the compatibility relation underlying a pip-space may always be coarsened, but not refined in general. There is an exception, however, namely the case of a scale of Hilbert spaces and analogous structures. We shall describe it in this section.

MR2681888El Harti, Rachid Extensions of σ-C∗-algebras. Operator algebras, operator theory and applications, 201–206, Oper. Theory Adv. Appl., 181, Birkhäuser Verlag, Basel, 2008. (Reviewer: Camillo Trapani)

?Almost? mean-field ising model: An algebraic approach

We study the thermodynamic limit of the algebraic dynamics for an "almost" mean-field Ising model, which is a slight generalization of the Ising model in the mean-field approximation. We prove that there exists a family of "relevant" states on which the algebraic dynamics αt can be defined. This αt defines a group of automorphisms of the algebra obtained by completing the standard spin algebra with respect to the quasiuniform topology defined by our states. © 1991 Plenum Publishing Corporation.

MR3035421 Samsonov, Boris F. Hermitian Hamiltonian equivalent to a given non-Hermitian one: manifestation of spectral singularity. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), no. 1989, 20120044, 11 pp. (Reviewer: Camillo Trapani)

Locally convex quasi *-algebras: basic aspects and commutative case

{$CQ\sp *$}-algebras: structure properties

Representable and Continuous Functionals on Banach Quasi *-Algebras

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.

Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues

Given a self-adjoint operator A in a Hilbert space H, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, i.e., for A to have a simple spectrum.

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.

Representable states on quasilocal quasi *-algebras

Continuing a previous analysis originally motivated by physics, we consider representable states on quasi-local quasi *-algebras, starting with examining the possibility for a {\em compatible} family of {\em local} states to give rise to a {\em global} state. Some properties of {\em local modifications} of representable states and some aspects of their asymptotic behavior are also considered.

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.

Partial inner product spaces: Theory and Applications

Partial Inner Product (PIP) Spaces are ubiquitous, e.g. Rigged Hilbert spaces, chains of Hilbert or Banach spaces (such as the Lebesgue spaces Lp over the real line), etc. In fact, most functional spaces used in (quantum) physics and in signal processing are of this type. The book contains a systematic analysis of PIP spaces and operators defined on them. Numerous examples are described in detail and a large bibliography is provided. Finally, the last chapters cover the many applications of PIP spaces in physics and in signal/image processing, respectively. As such, the book will be useful both for researchers in mathematics and practitioners of these disciplines.

Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals

The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra \({\mathfrak{A}_0}\) of a topological *-algebra \({\mathfrak{A}}\), satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given.

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.

Beyond frames: Semi-frames and reproducing pairs

Frames are nowadays a standard tool in many areas of mathematics, physics, and engineering. However, there are situations where it is difficult, even impossible, to design an appropriate frame. Thus there is room for generalizations, obtained by relaxing the constraints. A first case is that of semi-frames, in which one frame bound only is satisfied. Accordingly, one has to distinguish between upper and lower semi-frames. We will summarize this construction. Even more, one may get rid of both bounds, but then one needs two basic functions and one is led to the notion of reproducing pair. It turns out that every reproducing pair generates two Hilbert spaces, conjugate dual of each other. We …

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…

C*-seminorms on partial*-algebras: an overview

Locally convex quasi C*-algebras and noncommutative integration

In this paper we continue the analysis undertaken in a series of previous papers on structures arising as completions of C*-algebras under topologies coarser that their norm and we focus our attention on the so-called {\em locally convex quasi C*-algebras}. We show, in particular, that any strongly *-semisimple locally convex quasi C*-algebra $(\X,\Ao)$, can be represented in a class of noncommutative local $L^2$-spaces.

CQ*-Algebras of Operators and Application to Quantum Models

C*-algebras constitute, as known, the cornerstones of the Haag-Kastler [1] algebraic approach to quantum systems, with infinitely many degrees of freedom: one usually starts, in fact, with associating to each bounded region V of the configuration space of the system the C*-algebra A v of local observables in V. The uniform completion A of the algebra A0 generated by the Av’s is then considered as the C*-algebra of observables of the system.

Fully representable and*-semisimple topological partial*-algebras

We continue our study of topological partial *-algebras, focusing our attention to *-semisimple partial *-algebras, that is, those that possess a {multiplication core} and sufficiently many *-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous linear functionals and focus on the case where the two notions are completely interchangeable (fully representable partial *-algebras) with the scope of characterizing a *-semisimple partial *-algebra. Finally we describe various notions of bounded elements in such a partial *-algebra, in particular, those defined in terms of a positive cone (order bounded elements). The outcome …

Operators in Rigged Hilbert spaces: some spectral properties

A notion of resolvent set for an operator acting in a rigged Hilbert space $\D \subset \H\subset \D^\times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $\D$ and $\D^\times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.

*-Representations, seminorms and structure properties of normed quasi*-algebras

The class of -representations of a normed quasi -algebra (X;A0) is in- vestigated, mainly for its relationship with the structure of (X;A0). The starting point of this analysis is the construction of GNS-like -representations of a quasi -algebra (X;A0) dened by invariant positive sesquilinear forms. The family of bounded invariant positive sesquilinear forms denes some seminorms (in some cases, C -seminorms) that provide useful information on the structure of (X;A0) and on the continuity properties of its -representations. 1. Introduction. A quasi -algebra is a couple (X;A0), where X is a vector space with involution , A0 is a -algebra and a vector subspace of X, and X is an A0-bimodule who…

Locally Convex Quasi *-Algebras

This chapter is devoted to locally convex quasi *-algebras and locally convex quasi C*-algebras. Both these notions generalize what we have discussed in Chaps. 3 and 5. The advantage is, of course, that the range of applications becomes larger and larger; the drawback is that the theory becomes more involved.

PIP-Space Valued Reproducing Pairs of Measurable Functions

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ), where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

Lower Semi-frames, Frames, and Metric Operators

AbstractThis paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

Quasi *-algebras of measurable operators

Non-commutative $L^p$-spaces are shown to constitute examples of a class of Banach quasi *-algebras called CQ*-algebras. For $p\geq 2$ they are also proved to possess a {\em sufficient} family of bounded positive sesquilinear forms satisfying certain invariance properties. CQ *-algebras of measurable operators over a finite von Neumann algebra are also constructed and it is proven that any abstract CQ*-algebra $(\X,\Ao)$ possessing a sufficient family of bounded positive tracial sesquilinear forms can be represented as a CQ*-algebra of this type.

Spectral Properties of Partial *-Algebras

We continue our study of topological partial *algebras focusing our attention to some basic spectral properties. The special case of partial *-algebras of operators is examined first, in order to find sufficient hints for the study of the abstract case. The outcome consists in the selection of a class of topological partial *-algebras (partial GC*-algebras) that behave well from the spectral point of view and that allow, under certain conditions, a faithful realization as a partial O*-algebra.

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.

Banach elements and spectrum in Banach quasi *-algebras

A normal Banach quasi -algebra (X;A_0) has a distinguished Banach - algebra X_b consisting of bounded elements of X. The latter -algebra is shown to coincide with the set of elements of X having fi nite spectral radius. If the family P(X) of bounded invariant positive sesquilinear forms on X contains suffi ciently many elements then the Banach -algebra of bounded elements can be characterized via a C -seminorm defi ned by the elements of P(X).

TOPOLOGICAL PARTIAL *-ALGEBRAS: BASIC PROPERTIES AND EXAMPLES

Let [Formula: see text] be a partial *-algebra endowed with a topology τ that makes it into a locally convex topological vector space [Formula: see text]. Then [Formula: see text] is called a topological partial *-algebra if it satisfies a number of conditions, which all amount to require that the topology τ fits with the multiplier structure of [Formula: see text]. Besides the obvious cases of topological quasi *-algebras and CQ*-algebras, we examine several classes of potential topological partial *-algebras, either function spaces (lattices of Lp spaces on [0, 1] or on ℝ, amalgam spaces), or partial *-algebras of operators (operators on a partial inner product space, O*-algebras).

Auxiliary seminorms and the structure of a CQ*-algebra

After reviewing the main facts of the theory of CQ*-algebras, we give some new results on the structure of proper CQ*-algebras using some seminorms defined by certain families of positive sesquilinear forms.

Gibbs states, algebraic dynamics and generalized Riesz systems

In PT-quantum mechanics the generator of the dynamics of a physical system is not necessarily a self-adjoint Hamiltonian. It is now clear that this choice does not prevent to get a unitary time evolution and a real spectrum of the Hamiltonian, even if, most of the times, one is forced to deal with biorthogonal sets rather than with on orthonormal basis of eigenvectors. In this paper we consider some extended versions of the Heisenberg algebraic dynamics and we relate this analysis to some generalized version of Gibbs states and to their related KMS-like conditions. We also discuss some preliminary aspects of the Tomita-Takesaki theory in our context.

Some representation theorems for sesquilinear forms

The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$ defined on $\D$, is explored. The main result consists in showing that a sesquilinear form $\Omega$ is $\Theta$-regular, in the sense that it has a Radon-Nikodym type representation, if and only if it satisfies a sort Cauchy-Schwarz inequality whose right hand side is implemented by a positive sesquilinear form which is $\Theta$-absolutely continuous. In the particular case where $\Theta$ is an inner product in $\mathcal D$, this class of sesquilinear form cov…

A note on Banach partial *-algebras

MR3105981 81Q12 Znojil, Miloslav (CZ-AOS-N) The Coulomb potential and the paradoxes of PT symmetrization. J. Engrg. Math. 82 (2013), 173–185.