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Register a new userDuality for unbounded order convergence and applications
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Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) $X_{uo}^sim$ of a Banach lattice $X$ and identify it as the order continuous part of the order continuous dual $X_n^sim$. The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel-Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.
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The main result (roughly) is that if (H_i) converges weakly to H and if also f(H_i) converges weakly to f(H), for a single strictly convex continuous function f, then (H_i) must converge strongly to H. One application is that if f(pr(H)) = pr(f(H)), where pr denotes compression to a closed subspace M, then M must be invariant for H. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If h and f(h) are both quasimultipliers, then h must be a multiplier. Also (still roughly stated) if h and f(h) are both in pA_sa p, for a closed projection p, then h must be strongly q-continuous on p.
In 2012 Gu{a}vruc{t}a introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $mathcal{H}$, in order to decompose its range $mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:mathcal{D}(A)tomathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $finmathcal{D}(A)$ with respect to the norm of $mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $finmathcal{D}(A)$ with respect to the graph norm of $A$.
Let $d$ be a positive integer, and let $mu$ be a finite measure on $br^d$. In this paper we ask when it is possible to find a subset $Lambda$ in $br^d$ such that the corresponding complex exponential functions $e_lambda$ indexed by $Lambda$ are orthogonal and total in $L^2(mu)$. If this happens, we say that $(mu, Lambda)$ is a spectral pair. This is a Fourier duality, and the $x$-variable for the $L^2(mu)$-functions is one side in the duality, while the points in $Lambda$ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures $mu$ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in $br^d$; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.
We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.
In this paper we {em discuss} diverse aspects of mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize on the behaviour of row and column operators as they turn out to be the germs of an arbitrary matrix operator, providing most of the information about the latter {as it is the troublemaker}.
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