Noncommutative uncertainty principles

By Chunlan Jiang, Zhengwei Liu, Jinsong Wu

http://www.sciencedirect.com/science/article/pii/S0022123615003237

The classical uncertainty rules care for capabilities on abelian teams. during this paper, we speak about the uncertainty ideas for finite index subfactors which come with the circumstances for finite teams and finite dimensional Kac algebras. We end up the Hausdorff–Young inequality, Young's inequality, the Hirschman–Beckner uncertainty precept, the Donoho–Stark uncertainty precept. We symbolize the minimizers of the uncertainty ideas after which we turn out Hardy's uncertainty precept by utilizing minimizers. We additionally turn out that the minimizer is uniquely decided via the helps of itself and its Fourier remodel. The proofs take the good thing about the analytic and the categorial views of subfactor planar algebras. Our solution to turn out the uncertainty ideas additionally works for extra basic circumstances, similar to Popa's λ-lattices, modular tensor different types, and so forth.

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For a matrix A = (aij )i,j∈S in Spin 2,+ , tr 2 (A) = i∈S aii ; For a functionality f (i, j) = fij in Spin 2,− , tr 2 (f ) = n1 i,j∈S fij . The Fourier remodel F : Spin 2,± → Spin 2,∓ is F(A)(i, j) = √ fji naij ; F(f ) = ( √ )i,j∈S . n allow A = (aij ), B = (bij ) be n × n matrices. The Hadamard made of A and B is the matrix C = (cij ) (denoted through A ◦ B) given through cij = aij aij . The coproduct A ∗ B is √ discovered as nA ◦ B. C. Jiang et al. / magazine of practical research 270 (2016) 264–311 309 If there's a crew motion G on S, then the fixed element algebra of Spin less than the brought about team motion of G is usually a subfactor planar algebra, denoted through P .

Z. J. Math. , arXiv:math/9909027. [18] V. Jones, Index for subfactors, Invent. Math. seventy two (1983) 1–25. [19] V. Jones, Quadratic tangles in planar algebras, Duke Math. J. 161 (2012) 2257–2295. [20] V. Jones, D. Penneys, Infinite index subfactors and the GICAR different types, preprint, 2013. [21] H. Kosaki, functions of the advanced interpolation approach to a von Neumann algebra: noncommutative Lp -spaces, J. Funct. Anal. fifty six (1984) 29–78. C. Jiang et al. / magazine of sensible research 270 (2016) 264–311 311 [22] J. Kustermans, S.

Feel T : M → M is a linear map. If Tx p1 ≤ K1 x q1 and T x q1 ≤ K2 x q2 , then Tx the place 1 pθ = 1−θ p1 + θ 1 p2 , qθ = 1−θ q1 + pθ θ q2 , ≤ K11−θ K2θ x qθ , zero ≤ θ ≤ 1. facts. this can be a specified case of the interpolation theorem in [21]. ✷ Proposition four. 7. For any x in P2,± , we've F(x) ∞ ≤ x 1 tr 2 (|x|) = . δ δ facts. be aware that P2,± = I2,± ⊕ P2,± /I2,± = Ce1 ⊕ P2,± /I2,± . If x = e1 , F(x) = 1 and tr 2 (e1 ) = δ, then F(x) ∞ = 1 = tr 2 (|x|) . δ 278 C. Jiang et al. / magazine of useful research 270 (2016) 264–311 If x is a rank-one partial isometry v in P2,± /I2,± , then through Wenzl’s formulation (2), we've got δ v ≤ 1 , because the left part is a subprojection of the precise facet.

We are going to introduce a proposal of bi-shifts of biprojections by way of subfactor planar algebras as an an identical characterization of the minimizers. moreover, we are going to end up bi-shift of a biprojection is uniquely decided by means of the helps of itself and its Fourier rework. we want to work out the appliance of this bring about sign restoration. will probably be attention-grabbing to appreciate the uncertainty precept of infinite index subfactors. however the Fourier rework isn't transparent ordinarily. we want to consult [6,12,20] for a few recognized result of infinite index subfactors and [22] for in the neighborhood compact quantum teams.

Five in [25] whilst A is a finite abelian staff. As confirmed in [25], the minimizer of the classical uncertainty precept is C. Jiang et al. / magazine of practical research 270 (2016) 264–311 289 a nonzero scalar a number of of a translation and a modulation of the indicator functionality of a subgroup of A. Their ideas to explain the extremal bi-partial isometries don't paintings in subfactor planar algebras, seeing that we don't have the interpretation or the modulation to shift an extremal bi-partial isometry to a biprojection.

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