On the Grothendieck Theorem for jointly completely bounded bilinear forms

We show how the proof of the Grothendieck Theorem for jointly completely bounded bilinear forms on C*-algebras by Haagerup and Musat can be modified in such a way that the method of proof is essentially C*-algebraic. To this purpose, we use Cuntz algebras rather than type III factors. Furthermore, we show that the best constant in Blecher's inequality is strictly greater than one.Comment: 9 pages, minor change

On peak phenomena for non-commutative H∞H^\infty

A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\infty$-algebra $H^\infty(M,\tau)$ has unique predual,and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.Comment: final version (the presentation of some part is revised and one reference added

Multiplicativity of completely bounded p-norms implies a new additivity result

We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.Comment: Final version for Commun. Math. Physics. Section 5.2 of previous version deleted in view of the results in quant-ph/0601071 Other changes mino

Frobenius structures over Hilbert C*-modules

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.Comment: 35 page

Dual Operator Algebras with Normal Virtual h-Diagonal

International audienceWe study the class of dual operator algebras admitting a normal virtual h-diagonal (i.e. a diagonal in the normal Haagerup tensor product), this property can be seen as a dual operator space version of amenability. After giving several characterizations of these algebras, we show this class is stable under algebraic perturbations and cb-isomorphisms with small bound. We also prove some perturbation results for the Kadison-Kastler metric