Uffe Haagerup

Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one

The Fourier algebra A(G) of a locally compact group G is the space of matrix coefficients of the regular representation, and is the predual of the yon Neumann algebra VN(G) generated by the regular representation of G on L 2 (G). A multiplier m of A (G) is a bounded operator on A (G) given by pointwise multiplication by a function on G, also denoted m. We say m is a completely bounded multiplier ofA (G) if the transposed operator on VN(G) is completely bounded (definition below). It may be possible to find a net ofA (G)-functions, (m i : ie I) say, such that mi tends to

Operator valued weights in von Neumann algebras, II

Abstract An operator valued weight is a kind of generalized conditional expectation from a von Neumann algebra M to a sub von Neumann algebra N . If T is a n.f.s. (normal, faithful, semifinite) operator valued weight from M to N , and φ is a n.f.s. weight on N , then φ ° T defines a n.f.s. weight on M . Our main result is that σ t φ ° T extends σ t φ , and that the map φ → φ ° T preserves cocycle Radon Nikodym derivatives.

Random matrices with complex Gaussian entries

Abstract In this paper we give new and purely analytical proofs of a number of classical results on the asymptotic behavior of large random matrices of complex Wigner type (the GUE-case) or of complex Wishart type: Wigners semi-circle law, the Harer-Zagier recursion formula, the Marchenko-Pastur law, the Geman-Silverstein results on the largest and smallest eigenvalues and other related results. Our approach is based on the derivation of explicit formulae for the moment generating functions for random matrices of the two considered types.

The Grothendieck inequality for bilinear forms on C∗-algebras

Abstract The following generalization of Grothendiecks inequality is proved: For any bounded bilinear form V on a pair of C∗-algebras A, B, there exist two states ϕ1, ϕ2 on A and two states ψ1, ψ2 on B, such that |V(x,y)|⩽‖V‖(ϕ 1 (x ∗ x)+ϕ 2 (xx 2 )) 1 2 (φ 1 (y ∗ y)+ϕ 2 (yy 2 )) 1 2 for all xϵA and all yϵB. An inequality of this type was proved a few years ago by Pisier in the case where one of the C∗-algebras has the bounded approximation property. It follows from the above inequality that any bounded linear map T of a C∗-algebra into the dual of a C∗-algebra has a factorization T = R ∘ S through a Hilbert space, such that ‖ R ‖ ‖ S ‖ ⩽ 2 ‖ T ‖.

Equivalence of normal states on von Neumann algebras and the flow of weights

Abstract Let M be a von Neumann algebra. Two positive normal functionals ϕ, ψ on M are called equivalent, ϕ∼ψ, if ψ is in the norm-closure of the orbit of ϕ under the action of inner automorphisms. Our main result is an isometric characterization of the quotient space M + ∗ ∼ : We construct a natural isometry [ϕ] a \ g4 of M + ∗ ∼ into the set of positive normal functionals on “the smooth flow of weights” of M, where the smooth flow of weights is realized as the center Z(N) of the crossed product N = M × σω R for some faithful normal semifinite weight ω on M. As an application we obtain that an automorphism α on a factor M with separable predual acts trivially on M + ∗ ∼ if and only if α acts trivially on the smooth flow of weights, i.e., the Connes-Takesaki modulus mod(α) of α vanishes. We also obtain a new proof of the diameter formula diam (S n (M)/≈)=2 1 − λ 1 + λ for the quotient of the state space of a factor of type IIIλ, 0 ⩽ λ ⩽ 1.

A new upper bound for the complex Grothendieck constant

AbstractLet ϕ denote the real function

Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

\varphi (k) = k\smallint _0^{\pi /2} \frac{{cos^2 t}}{{\sqrt {1 - k^2 sin ^2 t} }}dt, - 1 \leqq k \leqq 1

The stable rank of some free product

and letKGC be the complex Grothendieck constant. It is proved thatKGC≦8/π(k0+1), wherek0 is the (unique) solution to the equationϕ(k)=1/8π(k+1) in the interval [0,1]. One has 8/π(k0+1) ≈ 1.40491. The previously known upper bound isKGC≦e1−y ≈ 1.52621 obtained by Pisier in 1976.