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Classification and GNS-construction for general universal products

Malte Gerhold

Institute for Mathematics and Informatics, Ernst-Moritz-Arndt University of Greifswald, Germany

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and

Stephanie Lachs

Institute for Mathematics and Informatics, Ernst-Moritz-Arndt University of Greifswald, Germany

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https://doi.org/10.1142/S0219025715500046Cited by:4 (Source: Crossref)

Abstract

It is known that there are exactly five natural products, which are universal products fulfilling two normalization conditions simultaneously. We classify universal products without these extra conditions. We find a two-parameter deformation of the Boolean product, which we call (r, s)-products. Our main result states that, besides degnerate cases, these are the only new universal products. Furthermore, we introduce a GNS-construction for not necessarily positive linear functionals on algebras and study the GNS-construction for (r, s)-product functionals.

Keywords:

AMSC: 46L53, 60A05, 81S25, 18D10

References

O. Arizmendi and T. Hasebe , Studia Math. 215 , 157 ( 2013 ) . Crossref, Web of Science, Google Scholar
O. Arizmendi and T. Hasebe , Proc. Amer. Math. Soc. 142 , 1621 ( 2014 ) . Crossref, Web of Science, Google Scholar
A. Ben Ghorbal and M. Schürmann , Math. Proc. Cambridge Philos. Soc. 133 , 531 ( 2002 ) . Crossref, Web of Science, Google Scholar
M. Bożejko , J. Reine Angew. Math. 377 , 170 ( 1987 ) . Web of Science, Google Scholar
M. Bożejko , M. Leinert and R. Speicher , Pacific J. Math. 175 , 357 ( 1996 ) . Crossref, Web of Science, Google Scholar
M. Bożejko and R. Speicher, Quantum Probability & Related Topics, QP–PQ VII (World Scientific, 1992) pp. 67–77. Link, Google Scholar
U. Franz, Quantum Independent Increment Processes. II, Lecture Notes in Math 1866 (Springer, 2006) pp. 161–257. Crossref, Google Scholar
U. Franz, R. Schott and M. Schürmann, Lévy processes and brownian motion on braided spaces, 2003 . Google Scholar
M. Gerhold, S. Kietzmann and S. Lachs, Noncommutative Harmonic Analysis with Applications to Probability III, Banach Center Publ 96 (Polish Acad. Sci. Inst. Math., Warsaw, 2012) pp. 175–191. Google Scholar
T. Hasebe , Colloq. Math. 124 , 79 ( 2011 ) . Crossref, Web of Science, Google Scholar
S. Lachs, Euler polynomials as central limits for universal products, Pre-print, 2014 . Google Scholar
S. Lachs, A new family of universal products and aspects of a non-positive quantum probability theory, Ph.D. thesis, EMAU Greifswald, 2015 . Google Scholar
R. Lenczewski, J. Funct. Anal. 258(12), 4075 (2010). Crossref, Web of Science, Google Scholar
S. Majid , Foundations of Quantum Group Theory ( Cambridge Univ. Press , 1995 ) . Crossref, Google Scholar
N. Muraki, On a q-deformation of free independence, talk at Matrix Aspects of Noncommutative Distributions and Free Probability . Google Scholar
N. Muraki, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6(3), 337 (2003). Link, Web of Science, Google Scholar
N. Muraki, Probab. Math. Statist. 33(2), 315 (2013). Web of Science, Google Scholar
M. Schürmann , White Noise on Bialgebras , Lecture Notes in Mathematics 1544 ( Springer , 1993 ) . Crossref, Google Scholar
R. Speicher, On universal products, in Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun., Vol. 12 (Amer. Math. Soc., 1997), pp. 257–266 . Google Scholar
R. Speicher and R. Woroudi, Boolean convolution, in Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun., Vol. 12 (Amer. Math. Soc., 1997), pp. 267–279 . Google Scholar
R. Street , Quantum Groups , Australian Mathematical Society Lecture Series 19 ( Cambridge Univ. Press , 2007 ) . Crossref, Google Scholar
D. Voiculescu, J. Operator Theory 17(1), 85 (1987). Web of Science, Google Scholar
W. von Waldenfels, Probability and Information Theory, II, Lecture Notes in Math 296 (Springer, 1973) pp. 19–69. Crossref, Google Scholar
J. Wysoczański, Noncommutative Harmonic Analysis with Applications to Probability, Banach Center Publ 78 (Polish Acad. Sci. Inst. Math., 2007) pp. 315–320. Crossref, Google Scholar