0000000001081789
AUTHOR
Salvatore Triolo
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.
Fredholm Spectra and Weyl Type Theorems for Drazin Invertible Operators
In this paper we investigate the relationship between some spectra originating from Fredholm theory of a Drazin invertible operator and its Drazin inverse, if this does exist. Moreover, we study the transmission of Weyl type theorems from a Drazin invertible operator R, to its Drazin inverse S.
Property (gab) through localized SVEP
In this article we study the property (gab) for a bounded linear operator T 2 L(X) on a Banach space X which is a stronger variant of Browder's theorem. We shall give several characterizations of property (gab). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gab) holds for large classes of operators and prove the stability of property (gab) under some commuting perturbations. 2010 Mathematics Subject Classication. Primary 47A10, 47A11; Secondary 47A53, 47A55.
Some perturbation results through localized SVEP
Some classical perturbation results on Fredholm theory are proved and extended by using the stability of the localized single-valued extension property under Riesz commuting perturbations. In the last part, we give some results concerning the stability of property (gR) and property (gb.
Radon-Nikodym theorem in quasi *-algebras
In this paper some properties of continuous representable linear functionals on a quasi $*$-algebra are investigated. Moreover we give properties of operators acting on a Hilbert algebra, whose role will reveal to be crucial for proving a Radon-Nikodym type theorem for positive linear functionals.
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 #.
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#.
Quasi-local quasi -algebras of measurable operators
In this paper we will continue the analysis undertaken in [1] and in [2] our investigation on the structure of Quasi-local quasi *-algebras. In this paper it is shown that any Quasi-local quasi -algebras (A;A_0), can be represented as a class of Banach C-modules called CQ-algebra of measurable operators in Segal's sense.
On Commuting Quasi-Nilpotent Operators that are Injective
Banach space operators that commute with an injective quasi-nilpotent operator, 11 such as the Volterra operator, inherit spectral and Fredholm properties, relating in 12 particular to the Weyl spectra.
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…
An invariant analytic orthonormalization procedure with applications
We apply the orthonormalization procedure previously introduced by two of us and adopted in connection with coherent states to Gabor frames and other examples. For instance, for Gabor frames we show how to construct $g(x)\in L^2(\Bbb{R})$ in such a way the functions $g_{\underline n}(x)=e^{ian_1x}g(x+an_2)$, $\underline n\in\Bbb{Z}^2$ and $a$ some positive real number, are mutually orthogonal. We discuss in some details the role of the lattice naturally associated to the procedure in this analysis.
Coordinate representation for non Hermitian position and momentum operators
In this paper we undertake an analysis of the eigenstates of two non self-adjoint operators $\hat q$ and $\hat p$ similar, in a suitable sense, to the self-adjoint position and momentum operators $\hat q_0$ and $\hat p_0$ usually adopted in ordinary quantum mechanics. In particular we discuss conditions for these eigenstates to be {\em biorthogonal distributions}, and we discuss few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with $\hat q$ and $\hat p$, based …
An invariant analytic orthonormalization procedure with an application to coherent states
We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators Aj, j=1,2,n, starting from a fixed normalized vector in H and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done. © 2007 American Institute of Physics.
Local Spectral Properties Under Conjugations
AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.
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.
Some Remarks on the Spectral Properties of Toeplitz Operators
In this paper, we study some local spectral properties of Toeplitz operators $$T_\phi $$ defined on Hardy spaces, as the localized single-valued extension property and the property of being hereditarily polaroid.
Local Spectral Theory for R and S Satisfying RnSRn = Rj
In this paper, we analyze local spectral properties of operators R,S and RS which satisfy the operator equations RnSRn=Rj and SnRSn=Sj for same integers j&ge
A Note on States and Traces from Biorthogonal Sets
In this paper, following Bagarello, Trapani, and myself, we generalize the Gibbs states and their related KMS-like conditions. We have assumed that H 0 , H are closed and, at least, densely defined, without giving information on the domain of these operators. The problem we address in this paper is therefore to find a dense domain D that allows us to generalize the states of Gibbs and take them in their natural environment i.e., defined in L &dagger
A note on semifinite von Neumann algebras
In this note we give some techniques for constructing a faithful semi finite trace on a semifinite von Neumann algebra.
Weyl-Type Theorems on Banach Spaces Under Compact Perturbations
In this paper, we study Browder-type and Weyl-type theorems for operators $$T+K$$ defined on a Banach space X, where K is (a non necessarily commuting) compact operator on X. In the last part, the theory is exemplified in the case of isometries, analytic Toeplitz operators, semi-shift operators, and weighted right shifts.
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…
Local spectral theory for Drazin invertible operators
Abstract In this paper we investigate the transmission of some local spectral properties from a bounded linear operator R, as SVEP, Dunford property (C), and property (β), to its Drazin inverse S, when this does exist.
A slight extension of the noncommutative integral.
In this paper we continue the analysis undertaken in [4]and in [12] on the general problem of extending positive linear functionals on a *-algebra and we construct a slight extension of the noncommutative integral.
Projections and isolated points of parts of the spectrum
In this paper, we relate the existence of certain projections, commuting with a bounded linear operator $T\in L(X)$ acting on Banach space $X$, with the generalized Kato decomposition of $T$. We also relate the existence of these projections with some properties of the quasi-nilpotent part $H_0(T)$ and the analytic core $K(T)$. Further results are given for the isolated points of some parts of the spectrum.
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.
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.
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.
Some invariant biorthogonal sets with an application to coherent states
We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-hermitian quantum mechanics.
Representations of Quasi–local quasi *–algebras and non–commutative integration
In this paper we are going to continue the analysis undertaken in [1] and [2] about the investigation on Quasi-local quasi *-algebras and their structure. Our aim is to show that any *-semisimple Quasi-local quasi *-algebra (A,A0) can be represented as a class of non-commutative L1-spaces.
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.
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.
Extensions of the Noncommutative Integration
In this paper we will continue the analysis undertaken in Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), Bongiorno et al. (Rocky Mt J Math 40(6):1745–1777, 2010), Triolo (Rend Circ Mat Palermo (2) 60(3):409–416, 2011) on the general problem of extending the noncommutative integration in a *-algebra of measurable operators. As in Aiena et al. (Filomat 28(2):263–273, 2014), Bagarello (Stud Math 172(3):289–305, 2006) and Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), the main problem is to represent different types of partial *-algebras into a *-algebra of measurable operators in Segal’s sense, provided that these partial *-algebras posses a sufficient family of pos…
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.
MODULI DI BANACH SU C*-ALGEBRE: Rappresentazioni Hilbertiane ed in spazi Lp non commutativi
La teoria delle *-rappresentazioni delle *-algebre localmente convesse o normate costituisce un argomento classico di cui dà conto una vasta letteratura. Le C*- algebre costituiscono sicuramente la classe di *-algebre di Banach per la quale la teoria delle rappresentazioni fornisce, probabilmente, i risultati più profondi ed importanti per le applicazioni. Nel 1964 R. Haag e D. Kastler formularono, in un celebre lavoro, il cosiddetto approccio algebrico alle teorie quantistiche per sistemi con infiniti gradi di libertà. In esso, ad una regione limitata dello spazio delle configurazioni del sistema, si associa la C*-algebra delle osservabili locali. L’unione di tutte queste C*-algebre costis…
CQ*-algebras and noncommutative measure
In this paper we continue the investigations in [4], [5], [8], [13], [14], [15], and [19], of the structure of quasi *-algebras and extend the results in [1] and [2]. Here, noncommutative Tp-spaces are shown to constitute examples of a class of Banach C*-modules called CQ*-algebras. Moreover, it is shown that any (strongly) *-semisimple proper CQ*-algebra (X ,A), with A a separable C*-algebra, can be represented as a CQ*-algebra of type Tp.
WQ*-algebras of measurable operators
Every C*-algebra \(\mathfrak{A}\) has a faithful *-representation π in a Hilbert space \(\mathcal{H}\). Consequently it is natural to pose the following question: under which conditions, the completion of a C*-algebra in a weaker than the given one topology, can be realized as a quasi *-algebra of operators? The present paper presents the possibility of extending the well known Gelfand — Naimark representation of C*-algebras to certain Banach C*-modules.
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.
Possible extensions of the noncommutative integral
In this paper we will discuss the problem of extending a trace σ defined on a dense von Neumann subalgebra \(\mathfrak{M}\) of a topological *-algebra \({\mathfrak{A}}\) to some subspaces of \({\mathfrak{A}}\). In particular, we will prove that extensions of the trace σ that go beyond the space L1(σ) really exist and we will explicitly construct one of these extensions. We will continue the analysis undertaken in Bongiorno et al. (Rocky Mt. J. Math. 40(6):1745–1777, 2010) on the general problem of extending positive linear functionals on a *-algebra.
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.