Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’

By Marius Junge. 2003

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We use Voiculescu’s concept of free probability to construct a completely isomorphic embedding of the operator space OH in the predual of a von Neumann algebra. We analyze the properties of this embedding and determine the operator space projection constant of OHn: $$\frac{1}{108}\sqrt{\frac{n}{1+\ln{n}}} \le \inf_{P:\mathcal{B}(\ell_2)\to{OH}_n, P^2=P} \left\|P\right\|_{cb} \le 288\pi\sqrt{\frac{2n}{1+\ln{n}}}.$$ The lower estimate is a recent result of Pisier and Shlyakhtenko that improves an estimate of order 1/(1+lnn) of the author. The additional factor $1 / \sqrt{1+\ln{n}}$ indicates that the operator space OHn behaves differently than its classical counterpart $\ell_2^n$. We give an application of this formula to positive sesquilinear forms on $\mathcal{B}(\ell_2)$. This leads to logarithmic characterization of C*-algebras with the weak expectation property introduced by Lance.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bd138a480afdbbc719582927e4f013566b0ed13f

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Citations in the corpus

A reduction method for noncommutative L_p-spaces and applications (2008, background): “The recent works on the complete embedding of Pisier’s operator Hilbert space OH into a noncommutative L1 in [Ju2] and certain Hilbertian homogeneous operator spaces into noncommutative Lp-spaces in [X3] show well how useful these Lp-spaces are as tools and examples for operator spaces.”
Rosenthal Type inequalities for free chaos. (2005, background): “The general case as stated above (or more generally using amalgamated free product) can be found in [14].”
Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory (2007, result): “Recent results in noncommutative harmonic analysis [25, 27, 28, 30] show the relevance of non-semifinite von Neumann algebras in the theory.”
Smooth Fourier multipliers on group von Neumann algebras (2010, background): “µ1) → BMO c S2 (Ω2)k. It is clear from the proofthat the little Grothendieck inequality becomes a crucial tool. Noncommutative forms of this result go back to Pisier’s work [73]. We know however from [40] that our argument above does not work for noncommutative algebras Mj. We refer the reader to [4”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, unknown): “Then Lp′(M; l1 )∗ = Lp(M; l∞) and Lp′(M; l∞) = Lp(M; l1 ) isometrically (see [J2] and [JX3]).”
Grothendieck's Theorem, past and present (2011, background): “The crucial case p = 1 was solved by Junge in [67] (see [130] for a simpler proof).”