Unbounded Operator Algebras and Representation Theory

Schmüdgen, Konrad

doi:10.1007/978-3-0348-7469-4

Cited by 397 publications

(450 citation statements)

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“…Due to the fact that for n > 1 not every nonnegative polynomial in R n can be written as a sum of squares of polynomials (see, for instance, [2, §6.3]), the moment problems in n variables are more difficult than the classical one variable problems. This very intriguing territory has been investigated by many authors (see [2], [7], [12] and their references), although characterizations for measures whose support lies in an arbitrary (generally unbounded) semi-algebraic set do not seem to exist.…”

Section: Introductionmentioning

confidence: 99%

Solving Moment Problems by Dimensional Extension

Putinar¹,

Vasilescu²

1999

The Annals of Mathematics

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Section: Introductionmentioning

confidence: 99%

Solving Moment Problems by Dimensional Extension

Putinar¹,

Vasilescu²

1999

The Annals of Mathematics

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“…For details and explicite example the reader is refered to [1 ; p. 270] and [14], respectively. The concept of *-representation ( [27]) is generalized to the one of J^-representation ( [25]). Along these lines it is implied that n<f> is a /-representation of j/.…”

Section: Ii) Iii)mentioning

confidence: 99%

“…From the mathematical point of view it is of interest to develop a general theory of unbounded representations of * -algebras, and there one has to deal with a class of representations which are not * -hermitean ( [27]). Typical examples of such representations are /-representations on Krein spaces (see § 3).…”

Section: § 1 Introductionmentioning

confidence: 99%

On the Cones of $α$- and Generalized $α$-Positivity for Quantum Field Theories with Indefinite Metric

Hofmann

1994

Publ. Res. Inst. Math. Sci.

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In order to construct a Krein-space theory (i.e., a *-algebra of (unbounded) operators which are defined on a common, dense, and invariant domain in a Krein space) the cones of a-positivity and generalized a-positivity are considered in tensor algebras. The relations between these cones, algebraic #-cones, and involutive cones are investigated in detail.Furthermore, an example of a P-functional 0 defined on (C 2 )® (tensor algebra over C 2 ) not being a-positive and yielding a non-trivial Krein-space theory is explicitely constructed. Thus, an affirmative answer to the question whether or not the method of P-functionals (introduced by Ota) is more general than the one of a-positivity (introduced by Jakobczyk) is provided in the case of tensor algebras. § 1. IntroductionThe present investigations are motivated by the algebraic approach to general (axiomatic) quantum field theory (QFT). The formalism introduced by Borchers [8] and Uhlmann [30] works for massive fields as well as for massless or gauge fields and is entirely equivalent to the Wightman axioms. Starting with a tensor algebra E® over some nuclear space E of test functions and a normalized continuous Hermitean functional W^E®' satisfying some further physically motivated conditions (linear program), a QFT is reconstructed by means of the GNS (Gelfand, Naimark, Segal) construction.It is now necessary to distinguish between i) massive fields, and ii) massless or gauge fields. In the case of i) W is taken to be a positive functional (Wightman functional), and thus the space of state vectors becomes a Hilbert space. In the case of ii) there are no-go theorems (see [22], [29]) stating that locality, covariance

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“…I, Chapter 3, Section 2]. Further different methods to solve the moment problem on nuclear spaces were introduced in 1975 in [16] and in [19] (see also [32] and [51,Section 12.5]). These approaches are essentially based on Choquet theory and decompositions of positive definite functionals on a commutative nuclear *-algebras into pure states.…”

Section: Introductionmentioning

confidence: 99%

Stochastic and Infinite Dimensional Analysis

Bernido¹,

Carpio-Bernido²,

Grothaus³

et al. 2016

Trends in Mathematics

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Abstract. This paper is aimed to show the essential role played by the theory of quasi-analytic functions in the study of the determinacy of the moment problem on finite and infinite-dimensional spaces. In particular, the quasianalytic criterion of self-adjointness of operators and their commutativity are crucial to establish whether or not a measure is uniquely determined by its moments. Our main goal is to point out that this is a common feature of the determinacy question in both the finite and the infinite-dimensional moment problem, by reviewing some of the most known determinacy results from this perspective. We also collect some properties of independent interest concerning the characterization of quasi-analytic classes associated to log-convex sequences.

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Unbounded Operator Algebras and Representation Theory

Cited by 397 publications

References 0 publications

Solving Moment Problems by Dimensional Extension

Solving Moment Problems by Dimensional Extension

On the Cones of $α$- and Generalized $α$-Positivity for Quantum Field Theories with Indefinite Metric

Stochastic and Infinite Dimensional Analysis

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