Operator algebra

Summary

In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory. Overview Operator algebras can be used to study arbitrary sets of operators with little algebraic relation simultaneously. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings. An operator algebra is typically required to be closed in a specified operator t

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https://en.wikipedia.org/wiki/Operator_algebra

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Geometric spects A of Similarity Problems

Peter Clemens Schlicht

This article presents a geometric approach to some similarity problems involving metric arguments in the non-positively curved space of positive invertible operators of an operator algebra and the canonical isometric action by invertible elements on the cone given by g . a = gag*. Through this approach, we extend and put into a geometric framework results by Pisier and partially answer a question by Andruchow et al. about minimality properties of canonical unitarizers.

OXFORD UNIV PRESS2018

GlobalBioIm: A Unifying Computational Framework for Solving Inverse Problems

Laurène Donati, Emmanuel Emilien Louis Soubies, Ferréol Arnaud Marie Soulez, Michaël Unser

We present a unifying framework for the development of state-of-the-art reconstruction algorithms in computational optics with a clear separation between the physical (forward model) and signal-related (regularization, incorporation of prior constraints) aspects of the problem. The pillars of our formulation are: (i) an operator algebra with its set of fast linear solvers, (ii) a variational derivation of reconstruction methods, and (iii) a suite of efficient numerical tools for the resolution of large-scale optimization problems. These core technologies are incorporated into a modular software library featuring the key components for the implementation and testing of iterative reconstruction algorithms. The concept is illustrated with concrete examples in 3D deconvolution microscopy and lenseless imaging.

OSA2017

From Scale Invariance to Lorentz Symmetry

Sergey Sibiryakov

It is shown that a unitary translationally invariant field theory in 1 + 1 dimensions, satisfying isotropic scale invariance, standard assumptions about the spectrum of states and operators, and the requirement that signals propagate with finite velocity, possesses an infinite dimensional symmetry given by one or a product of several copies of conformal algebra. In particular, this implies the presence of one or several Lorentz groups acting on the operator algebra of the theory.

Amer Physical Soc2014