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HEREDITARILY ANTISYMMETRIC OPERATOR ALGEBRAS

Part of: Topological (rings and) algebras with an involution Ordered sets

Published online by Cambridge University Press:  09 October 2019

Nik Weaver Open the ORCID record for Nik Weaver [Opens in a new window]
Nik Weaver*
Affiliation:
Department of Mathematics, Washington University in Saint Louis, Saint Louis, MO63130, USA (nweaver@math.wustl.edu)
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Abstract

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We introduce a notion of ‘hereditarily antisymmetric’ operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case.

Keywords

antisymmetric operator algebrasquantum posets

MSC classification

Primary: 46K50: Nonselfadjoint (sub)algebras in algebras with involution
Secondary: 06A75: Generalizations of ordered sets
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 1039 - 1074
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Copyright
© Cambridge University Press 2019

References

Davidson, K., Nest Algebras, Pitman Research Notes in Mathematical Sciences, Volume 191 (Longman Scientific and Technical, Marlow, 1988), 411 pp.Google Scholar
Day, M. M., Means for the bounded functions and ergodicity of the bounded representations of semigroups, Trans. Amer. Math. Soc. 69 (1950), 276–291.CrossRefGoogle Scholar
Dixmier, J., Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. (Szeged) 12 (1950), 213–227.Google Scholar
Duan, R., Severini, S. and Winter, A., Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number, IEEE Trans. Inform. Theory 59 (2013), 1164–1174.CrossRefGoogle Scholar
Kastis, E. and Power, S. C., The operator algebra generated by the translation, dilation and multiplication semigroups, J. Funct. Anal. 269 (2015), 3316–3335.CrossRefGoogle Scholar
Katavolos, A. and Power, S. C., Translation and dilation invariant subspaces of L 2(ℝ), J. Reine Angew. Math. 552 (2002), 101–129.Google Scholar
Kuperberg, G. and Weaver, N., A von Neumann algebra approach to quantum metrics, Mem. Amer. Math. Soc. 215 (2012), v, 1–80.Google Scholar
Radjavi, H. and Rosenthal, P., Simultaneous Triangularization (Springer-Verlag, New York, 2000), 318 pp.CrossRefGoogle Scholar
Wacław, S., Antisymmetric operator algebras I, II, Ann. Polon. Math. 37 (1980), 263–274, 299–311.Google Scholar
Weaver, N., Quantum relations, Mem. Amer. Math. Soc. 215 (2012), v–vi, 81–140.Google Scholar
Weaver, N., A ‘quantum’ Ramsey theorem for operator systems, Proc. Amer. Math. Soc. 145 (2017), 4595–4605.CrossRefGoogle Scholar
Weaver, N., The ‘quantum’ Turan problem for operator systems, Pacific J. Math. 301 (2019), 335–349.CrossRefGoogle Scholar
Weaver, N., Quantum graphs as quantum relations, manuscript.Google Scholar