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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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28 Feb 2011TL;DR: In this paper, the authors develop the necessary dilation theory for both models, which determines the C*-envelope of the tensor algebra and the conjugacy operator algebras for the universal algebra of the system.
Abstract: Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X,\tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.|Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\sigma_i:X \to X$ for $1 \le i \le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\mathcal{A}(X,\tau)$ and the semicrossed product $\mathrm{C}_0(X)\times_\tau\mathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.
66 citations
TL;DR: In this article, it was shown that the Frobenius-optimal preconditioners can be used to define new and just known linear positive operators uniformly approximating the function f. The convergence features of these PCG have been naturally studied by means of the Weierstrass-Jackson Theorem, owing to the profound relationship between the spectral features of the matrices generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself.
Abstract: Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system \(A_n\vec{x}=\vec{b}\) [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from \(A_n\). The convergence features of these PCG have been naturally studied by means of the Weierstrass–Jackson Theorem [17,36,45], owing to the profound relationship between the spectral features of the matrices \(A_n\), generated by the Fourier coefficients of a continuous function f, and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f. On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense.
66 citations
TL;DR: For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate.
Abstract: One of von Neumann's motivations for developing the theory of operator algebras and his and Murray's 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type ${\rm III}_{1}$ factor in Connes' classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other.
66 citations
TL;DR: In this paper, the authors initiated the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator-algebras, and they initiated the work of generalized generalized Jordan triples.
Abstract: In this paper we initiate the study of generalized Jordan derivations
and generalized Jordan triple derivations on prime rings and standard operator
algebras.
65 citations
TL;DR: In this article, the authors use continuous model theory to obtain several results concerning isomorphisms and embeddings between II-1 factors and their ultrapowers, including a poor man's resolution of the Connes embedding problem.
Abstract: We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.
65 citations