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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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TL;DR: In this article, the symmetric operators of the q-deformed Heisenberg algebra were extended to essentially self-adjoint operators, and the respective diagonalization can be achieved by aqdeformed Fourier transformation.
Abstract: We extend the symmetric operators of theq-deformed Heisenberg algebra to essentially self-adjoint operators. On the extended domains the product of the operators is not defined. To represent the algebra we had to enlarge the representation and we find a Hilbert space representation of the deformed Heisenberg algebra in terms of essentially self-adjoint operators. The respective diagonalization can be achieved by aq-deformed Fourier transformation.
71 citations
TL;DR: In this article, the closed faces in twelve kinds of convex sets that appear in operator algebra theory are described, including positive parts of unit balls for C*-algebras and their dual spaces.
Abstract: We give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory. These consist of positive parts of unit balls for C*-algebras and their dual spaces, and for von Neumann algebras and their pre-duals; of self-adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes. All these faces are shown to be semi-exposed and naturally paired with a polar face in the dual (or pre-dual) space
71 citations
TL;DR: In this paper, the authors consider local field theory on κ-deformed Minkowski space, which is an example of solvable Lie-algebraic noncommutative structure.
Abstract: We consider local field theory on κ-deformed Minkowski space which is an example of solvable Lie-algebraic noncommutative structure. Using integration formula over κ-Minkowski space and κ-deformed Fourier transform, we consider for deformed local fields the reality conditions as well as deformation of action functionals in standard Minkowski space. We present explicit formulas for two equivalent star products describing CBH quantization of field theory on κ-Minkowski space. We express also via star product technique the noncommutative translations in κ-Minkowski space by commutative translations in standard Minkowski space.
70 citations
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TL;DR: In this paper, the notion of L^2 homology and Betti numbers for a tracial von Neumann algebra with a trace was defined and a link between the first L 2 Betti number and free entropy dimension was found.
Abstract: We define the notion of L^2 homology and L^2 Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L^2 homology for groups, using the ideas from the theory of correspondences. For the group algebra of a discrete group, our Betti numbers coincide with the L^2 Betti numbers of the group. We find a link between the first L^2 Betti number and free entropy dimension, which points to the non-vanishing of L^2 homology for the von Neumann algebra of a free group.
70 citations
TL;DR: For simple unweighted shift operators, a family of complex eigenvalue eigenstates of the shift down operators, called theharmonious states, is constructed in this article, where every density matrix is realized as a weighted sum of projections to the harmonious states.
Abstract: For simple unweighted shift operators a family of complex eigenvalue eigenstates of the shift down operators, called theharmonious states, is constructed. Every density matrix is realized as a weighted sum of projections to the harmonious states; and the weight distributions serve as quasiprobability densities for normal ordered operators.
70 citations