. A collection of several different probabilistic processes involving trees is shown to have an unexpected unity. This makes possible a fruitful interplay of these probabilistic processes. The processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well. Our work has five specific aims: First, an exact correspondence between random walks and percolation on trees is proved, extending and sharpening previous work of the author. This is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes. Second, we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation. This capacitary formulation extends and sharpens work of Fan on random interval coverings. Third, we show how this formulation also applies to generalize work of Evans on random labelling of trees. Fourth, the correspondence between random walks and percolation is used...

this paper, we show that the answer to the above question is positive. More specifically, we give a canonical construction for semi-pullbacks in the category whose objects are families of Markov processes, with given transition kernels, on Polish spaces and whose morphisms are transition probability preserving surjective continuous maps. One immediate consequence is that the category of probability measures on Polish spaces with measure-preserving continuous maps has semi-pullbacks. Our construction gives semi-pullbacks for various full subcategories, including that of Markov processes on locally compact second countable spaces and also in the larger category where the objects are Markov processes on analytic spaces (i.e. continuous images of Polish spaces) and morphisms are transition probability preserving surjective Borel maps. It also applies to the corresponding categories of ultrametric spaces. Finally, our result also holds in the larger categories with Markov processes which are given by subprobability distributions, i.e. the total probability of transition from a state can be strictly less than one. We now explain the relevance of our result in computer science. The consequences of Semi-pullbacks and Bisimulation 3 our mathematical result in the theory of probabilistic bisimulation has been investigated in (Blute et al., 1997; Desharnais et al., 1998). We will briefly review this here. Following the work of Joyal, Nielsen and Winskel (Joyal et al., 1996) on the notion of bisimulation using open maps, define two objects A and B in a category to be bisimular if there exists an object C and morphisms f : C ! A and g : C ! B, i.e.,

Abstract. An analysis is given of ∗-representations of rank 2 single vertex graphs. We develop dilation theory for the nonselfadjoint algebras Aθ and Au which are associated with the commutation relation permutation θ of a 2 graph and, more generally, with commutation relations determined by a unitary matrix u in Mm(C)⊗Mn(C). We show that a defect free row contractive representation has a unique minimal dilation to a ∗-representation and we provide a new simpler proof of Solel’s row isometric dilation of two u-commuting row contractions. Furthermore it is shown that the C*-envelope of Au is the generalised Cuntz algebra OXu for the product system Xu of u; that for m ≥ 2 and n ≥ 2 contractive representations of Aθ need not be completely contractive; and that the universal tensor algebra T+(Xu) need not be isometrically isomorphic to Au. 1.

Abstract. This paper lays down a foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theory based on the representation theory of SL 2 (R) group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types.

Abstract. We provide a detailed analysis of atomic ∗-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. 1.

Abstract. B. Magajna and J. Schweizer showed in 1997 and 1999, respectively, that C*-algebras of compact operators can be characterized by the property that every norm-closed (and coinciding with its biorthogonal complement, resp.) submodule of every Hilbert C*-module over them is automatically an orthogonal summand. We find out further generic properties of the category of Hilbert C*-modules over C*-algebras which characterize precisely the C*-algebras of compact operators. In 1997 B. Magajna obtained the equivalence of the property of the category of Hilbert C*-modules over a certain C*-algebra A that any Hilbert C*-submodule is automatically an orthogonal summand with the property of the C*-algebra A of coefficients to admit a faithful ∗-representation in some C*-algebra of compact operators on some Hilbert space, cf. Theorem 2.1. In 1999 J. Schweizer was able to sharpen the argument replacing the Hilbert C*-module property of B. Magajna by the property of the category of Hilbert C*-modules over a certain C*-algebra A that any Hilbert C*-submodule which coincides with its biorthogonal complement is automatically an orthogonal summand, cf. Theorem 2.1. Later on in 2003 M. Kusuda published further results which indicate that in the majority of situations the Hilbert C*-module property can be weakened merely requiring the KA(M)-Asubbimodules of the Hilbert C*-modules M to be always orthogonal summands, see [11, 12] for the details. Studying the work of B. Magajna and J. Schweizer C*-algebras A of the form A = c0- ∑ α ⊕K(Hα) become of special interest, where the symbol K(Hα) denotes the C*-algebra of all compact operators on some Hilbert space Hi, and the c0-sum is either a finite block-diagonal sum or a block-diagonal sum with a c0-convergence condition on the C*-algebra

Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing us to develop a dilation theory for higher rank operator tuples. These dilations are joint dilations of the families of operators satisfying relations encoded by the graph structure which we call Λ-contractions or Λ-isometries. Besides commutant lifting results and characterisations of pure states on higher rank graph algebras several applications to the structure theory of non-selfadjoint graph operator algebras are presented generalising recent results in special cases.

Abstract. Let X be a locally compact Hausdorff space with n proper continuous self maps τi: X → X for 1 ≤ i ≤ n. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra A(X, τ) and the semicrossed product C0(X) ×τ F + n. We introduce a concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from either the algebraic structure of A(X, τ) or C0(X) ×τ F + n. Various classification results follow as a consequence. For example, for n = 2, 3, the tensor algebras are (algebraically or even completely isometrically) isomorphic if and only if the systems are piecewise topologically conjugate. In order to establish these results we make use of analytic varieties as well as homotopy theory for Lie groups We define a generalized notion of wandering sets and recurrence.

Abstract. We give a new very concrete description of the C*envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal. 1.

Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with a environment that is “small ” in the sense dE ≤ dB. Such channels include