We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on Mn(C) for all n ≥ 3 , as well as an example of a one-parameter semigroup (T (t)) t≥0 of Markov maps on M 4 (C) such that T (t) fails to be factorizable for all small values of t > 0 . As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality. a unique completely positive, unital map T * : N → M such thatIt is easy to show that T * is a (ψ , φ)-Markov map. Remark 1.2. A special case of interest is the one of a (φ, ψ)-Markov map J : M → N which is a *monomorphism. In this case J(M ) is a σ ψ -invariant sub-von Neumann algebra of N . Hence, by [34], there is a unique ψ-preserving normal faithful conditional expectation E J(M) of N onto J(M ), and thus the adjoint J * is given by J * = J −1 • E J(M) . C. Anantharaman-Delaroche proved in [2] that the noncommutative analogue of Rota's theorem holds for Markov maps which are factorizable in the following sense (cf. Definition 6.2 in [2]): Definition 1.3. A (φ, ψ)-Markov map T : M → N is called factorizable if there exists a von Neumann algebra P equipped with a faithful normal state χ , and * -monomorphisms J 0 : M → P and J 1 : N → P such that J 0 is (φ, χ)-Markov and J 1 is (ψ, χ)-Markov, satisfying, moreover, T = J * 0 • J 1 .

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Abstract. In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C * -algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact. Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C * -algebras, of which at least one is exact. In this paper we prove that the Effros-Ruan conjecture holds for general C * -algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear form on a pair of C * -algebras A and B , there exist states f 1 , f 2 on A and g 1 , g 2 on B such that for all a ∈ A and b ∈ B ,While the approach by Pisier and Shlyakhtenko relies on free probability techniques, our proof uses more classical operator algebra theory, namely, Tomita-Takesaki theory and special properties of the Powers factors of type III λ , 0 < λ < 1 .

By analogy with the well-established notions of just-infinite groups and just-infinite (abstract) algebras, we initiate a systematic study of just-infinite C˚-algebras, i.e., infinite dimensional C˚-algebras for which all proper quotients are finite dimensional. We give a classification of such C˚-algebras in terms of their primitive ideal space, that leads to a trichotomy. We show that just-infinite, residually finite dimensional C˚-algebras do exist by giving an explicit example of (the Bratteli diagram of) an AF-algebra with these properties.Further, we discuss when C˚-algebras and˚-algebras associated with a discrete group are just-infinite. If G is the Burnside-type group of intermediate growth discovered by the first-named author, which is known to be just-infinite, then its group algebra CrGs and its group C˚-algebra C˚pGq are not just-infinite. Furthermore, we show that the algebra B " πpCrGsq under the Koopman representation π of G associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional C˚-algebra Cπ pGq is just-infinite.As we shall later describe just-infinite C˚-algebras in terms of their primitive ideal space, and as the interesting cases of just-infinite C˚-algebras are those that are residually finite dimensional, we review in this section the relevant background. The primitive ideal space of a C˚-algebraA C˚-algebra A is said to be primitive if it admits a faithful irreducible representation on some Hilbert space. It is said to be prime if, whenever I and J are closed two-sided ideals in A such that I X J " 0, then either I " 0, or J " 0. It is easy to see that every primitive C˚-algebra is prime, and it is a non-trivial result that the converse holds for all separable C˚-algebras; cf. [31, Proposition 4.3.6]. However, there are (complicated) examples of non-separable C˚-algebras that are prime, but not primitive, see [37].A closed two-sided ideal I in a C˚-algebra A is said to be primitive if I ‰ A and I is the kernel of an irreducible representation of A on some Hilbert space. The primitive ideal space, PrimpAq, is the set of all primitive ideals in A. A closed two-sided ideal I of A is primitive if and only if the quotient A{I is a primitive C˚-algebra. In particular, 0 P PrimpAq if and only if A is primitive. The primitive ideal space is a T 0 -space when equipped with the hull-kernel topology, which is given as follows: the closure F of a subset F Ď PrimpAq consists of all ideals I P PrimpAq which contain Ş JPF J. If A is primitive, so that 0 P PrimpAq, then t0u " PrimpAq. In the commutative case, the primitive ideal space is the usual spectrum: PrimpC 0 pXqq is homeomorphic to X, whenever X is a locally compact Hausdorff space. The following fact will be used several times in the sequel:Remark 2.1. Each finite dimensional C˚-algebra A is (isomorphic to) a direct sum of full matrix algebras,for some positive integers n, k 1 , k 2 , . . . , k n . As each matrix algebra is simple, PrimpAq can be naturally...

Abstract. We prove a noncommutative version of the John-Nirenberg theorem for nontracial filtrations of von Neumann algebras. As an application, we obtain an analogue of the classical large deviation inequality for elements of the associated BM O space.