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Levy area for the free Brownian motion: existence and non-existence. (English) Zbl 1062.46055
The author shows that the Lévy area of the piecewise linear path which coincides with \((X_s)_{0\leq s\leq 1}\) (free Brownian motions) at the points of a subdivision \(D\) tends to the Lévy area of the free Brownian motion when the mesh size of \(D\) tends to \(0\). To this end, the author proves a Burkholder-Davis-Gundy type inequality. Then he shows that there exists no multiplicative functional above the free Brownian motion, that is, in other words, there does not exist a Lévy area in the projective tensor product.

46L54 Free probability and free operator algebras
Full Text: DOI
[1] Capitaine, M.; Donati-Martin, C., The Lévy area process for the free Brownian motion, J. funct. anal., 179, 1, 153-169, (2001) · Zbl 0979.60044
[2] Lyons, T., Differential equations driven by rough signals, Rev. mat. iberoamericana, 14, 2, 215-310, (1998) · Zbl 0923.34056
[3] Pisier, G.; Xu, Q., Non-commutative martingales inequalities, Commun. math. phys., 189, 667-698, (1997) · Zbl 0898.46056
[4] Takesaki, M., Theory of operator algebras, I, (1979), Springer New York, Heidelberg · Zbl 0990.46034
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