The Covariant Stone–von Neumann Theorem for Actions of Abelian Groups on $$ C^{*} $$-Algebras of Compact Operators

By proving various equivalent versions of the generalized Weyl-von Neumann theorem, we investigate the structure of projections in the multiplier algebra [Formula: see text] of certain C*-algebra [Formula: see text] with real rank zero. For example, we prove that [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. In case [Formula: see text] is a purely infinite simple C*-algebra, [Formula: see text] if and only if any two projections in [Formula: see text] are simultaneously quasidiagonal. If [Formula: see text] is one of the Cuntz algebras, or one of finite factors or type III factors, then any two projections in [Formula: see text] are simultaneously quasidiagonal. On the other hand, if [Formula: see text] is one of the Bunce-Deddens algebras or one of the irrational rotation algebras of real rank zero, then there exist two projections in [Formula: see text] which are not simultaneously quasidiagonal.

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A von Neumann theorem for uniformly distributed sequences of partitions

Rendiconti del Circolo Matematico di Palermo (1952 -) ◽

10.1007/s12215-011-0030-x ◽

2011 ◽

Vol 60 (1-2) ◽

pp. 83-88 ◽

Cited By ~ 4

Author(s):

Ingrid Carbone ◽

Aljosa Volcic

Keyword(s):

Von Neumann ◽

Neumann Theorem ◽

Uniformly Distributed Sequences ◽

Von Neumann Theorem

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When does the Weyl-von Neumann Theorem hold?

Bulletin of the London Mathematical Society ◽

10.1112/blms.12064 ◽

2017 ◽

Vol 49 (4) ◽

pp. 742-744

Author(s):

Hiroshi Ando ◽

Yasumichi Matsuzawa

Keyword(s):

Von Neumann ◽

Neumann Theorem ◽

Von Neumann Theorem

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Superunitary Representations of Heisenberg Supergroups

International Mathematics Research Notices ◽

10.1093/imrn/rny184 ◽

2018 ◽

Vol 2020 (19) ◽

pp. 5926-6006 ◽

Cited By ~ 1

Author(s):

Axel de Goursac ◽

Jean-Philippe Michel

Keyword(s):

Representation Theory ◽

Heisenberg Group ◽

Fourier Transformation ◽

Coadjoint Orbits ◽

New Approach ◽

Von Neumann ◽

Neumann Theorem ◽

Lie Supergroups ◽

Definition Of ◽

Von Neumann Theorem

Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

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FUNDAMENTAL SYMMETRIES OF THE EXTENDED SPACETIME

International Journal of Modern Physics A ◽

10.1142/s0217751x08040196 ◽

2008 ◽

Vol 23 (08) ◽

pp. 1266-1269

Author(s):

ORCHIDEA MARIA LECIAN ◽

GIOVANNI MONTANI

Keyword(s):

Uncertainty Relations ◽

Fundamental Symmetries ◽

Von Neumann ◽

Neumann Theorem ◽

Von Neumann Theorem

On the basis of Fourier duality and Stone-von Neumann theorem, we will examine polymer-quantization techniques and modified uncertainty relations as possible 1-extraD compactification schemes for a phenomenological truncation of the extraD tower.

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