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In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, that allows us to express the Markov property for the local structure of the graph. Our main result concerns the existence and uniqueness of quantum Markov field over such graphs.
Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital $*$-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, s
We study the range of a classifiable class ${cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${cal A}$ contains all unital simple AH-algebras. We show that all unital simple induct
This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on loc
Suppose that $X_{1}$ and $X_{2}$ are two selfadjoint random variables that are freely independent over an operator algebra $mathcal{B}$. We describe the possible operator atoms of the distribution of $X_{1}+X_{2}$ and, using linearization, we determi
We introduce the notion of additive units, or `addits, of a pointed Arveson system, and demonstrate their usefulness through several applications. By a pointed Arveson system we mean a spatial Arveson system with a fixed normalised reference unit. We