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The complex Grothendieck inequality for 2\(\times 2\) matrices. (English) Zbl 0661.47011

Grothendieck’s inequality asserts that there is an absolute constant K such that \[ | \sum^{n}_{i,j=1}a_{ij}<x_ i,y_ j>| \leq K\| a\|_{\infty,1}, \] where \(x_ 1,x_ 2,...,x_ n\), \(y_ 1,y_ 2,...,y_ n\) are elements of the unit ball of a Hilbert space and \[ \| a\|_{\infty,1}=\sup \{| \sum^{n}_{i,j=1}a_{ij}s_ it_ j|:\quad | s_ i| \leq 1,\quad | t_ i| \leq 1\}. \] The paper contains a proof of Davie’s result: If \(n=2\), the best value of K is \(K=1\).
Reviewer: L.A.Sakhnovich

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46M05 Tensor products in functional analysis
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