Faithfulness of Free Product States

Faithfulness of Free Product States

It is proved that the free product state, in the reduced free product of C–algebras, is faithful if the states one started with are faithful. The reduced free product of C–algebras was introduced by Voiculescu as the appropriate construction for C–algebras in the setting of his theory of freeness [1]. (See also the book [2]). Given a set I and for each ι ∈ I a unital C–algebra Aι with a state φ...

On the incompatibility of faithfulness and monotone DAG faithfulness

Cheng, Greiner, Kelly, Bell and Liu [Artificial Intelligence 137 (2002) 43–90] describe an algorithm for learning Bayesian networks that—in a domain consisting of n variables—identifies the optimal solution using O(n4) calls to a mutual-information oracle. This result relies on (1) the standard assumption that the generative distribution is Markov and faithful to some directed acyclic graph (DA...

On the Nilpotent Multiplier of a Free Product

The Emergence of Faithfulness

Any theory of language development, whether emergentist or nativist, must address the child's phonological development. Children's pronunciations of words are often quite different from those of adults. The child pronunciations may diverge notably from the target adult phonology, but in ways that make sense from a cross-linguistic perspective on the phonological systems of adult languages. In t...

Free Quasi-free States

To a real Hilbert space and a one-parameter group of orthogonal transformations we associate a C∗-algebra which admits a free quasi-free state. This construction is a freeprobability analog of the construction of quasi-free states on the CAR and CCR algebras. We show that under certain conditions, our C∗-algebras are simple, and the free quasi-free states are unique. The corresponding von Neuma...