Abstract. Consider a completely bounded Fourier multiplier φ of a locally compact group G, and take 1 ≤ p ≤ ∞. One can associate to φ a Schur multiplier on the Schatten classes Sp(L 2 G), as well as a Fourier multiplier on L p (LG), the non-commutative L p -space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the L p -Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups.For a discrete group G and in the special case when p = 2 is an even integer, we show the following. If there exists a map between L p (LG) and an ultraproduct of L p (M) ⊗ Sp(L 2 G) that intertwines the Fourier multiplier with the Schur multiplier, then G must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra M n (C) of complex n × n matrices. This leads to an explicit expression for the pointfree quantum phase space n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the KochenSpecker Theorem.In our approach, the nondistributive lattice P(M n (C)) of projections in M n (C) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O( n ) of functions from the poset C(M n (C)) of all unital commutative C*-subalgebras C of M n (C) to P(M n (C)). The lattice O( n ) is essentially the (pointfree) topology of the quantum phase space n , Dedicated to Pekka Lahti, at his 60th birthday.M. Caspers ( ) · C. Heunen · N.P. Landsman 732 Found Phys (2009) 39: 731-759 and as such defines a Heyting algebra. Each element of O( n ) corresponds to a "Bohrified" proposition, in the sense that to each classical context C ∈ C(M n (C)) it associates a yes-no question (i.e. an element of the Boolean lattice P(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.Keywords Quantum logic · Topos theory · Intuitionistic logic 'All departures from common language and ordinary logic are entirely avoided by reserving the word "phenomenon" solely for reference to unambiguously communicable information, in the account of which the word "measurement" is used in its plain meaning of standardized comparison. ' (N. Bohr [4])

Abstract. Using interpolation properties of non-commutative L p -spaces associated with an arbitrary von Neumann algebra, we define a L p -Fourier transform 1 ≤ p ≤ 2 on locally compact quantum groups. We show that the Fourier transform determines a distinguished choice for the interpolation parameter as introduced by Izumi. We define a convolution product in the L p -setting and show that the Fourier transform turns the convolution product into a product. IntroductionThe Fourier transform is one of the most powerful tools coming from abstract harmonic analysis. Many classical applications, in particular in the direction of L p -spaces, can be found in for example [6]. Here we extend this tool by giving a definition of a Fourier transform on the non-commutative L p -spaces associated with a locally compact quantum group. This gives a link between quantum groups and non-commutative measure theory.Recall that the Fourier transform on locally compact abelian groups can be defined in an L p -setting for p any real number between 1 and 2. This is done in the following way. Let G be a locally compact group and letĜ be its Pontrjagin dual. For a L 1 -function f on G, we define its Fourier transformf to be the function onĜ, which is defined byThenf is a continuous function onĜ vanishing at infinity. So we can consider this transform as a bounded mapthenf is a L 2 -function onĜ and this map extends to a unitary map. It is known that the Fourier transform can be generalized in a L p -setting by means of the Riesz-Thorin theorem, see [1]. The statement of this theorem directly implies the following. For any p, with 1 ≤ p ≤ 2, the linear mapf →f extends uniquely to a bounded mapThis map F p is known as the L p -Fourier transform.Date: May 10th, 2011.

Let M be a semi-finite von Neumann algebra and let f :. This result resolves a conjecture raised by F. Nazarov and V. Peller implying a couple of existing results in perturbation theory.

Abstract:The Haagerup approximation property for a von Neumann algebra equipped with a faithful normal state ϕ is shown to imply existence of unital, ϕ-preserving and KMS-symmetric approximating maps. This is used to obtain a characterisation of the Haagerup approximation property via quantum Markov semigroups (extending the tracial case result due to Jolissaint and Martin) and further via quantum Dirichlet forms.