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Register a new userLimit algebras and integer-valued cocycles, revisited
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A triangular limit algebra A is isometrically isomorphic to the tensor algebra of a C*-correspondence if and only if its fundamental relation R(A) is a tree admitting a $Z^+_0$-valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining C*-correspondence and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho-Hopcroft-Ullman algorithm from computer aided graph theory.
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This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E_0-semigroup. The gauge group of a spatial E_0-semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E_0-semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k, then the gauge group action is transitive on the set of k+1-tuples of appropriately normalized independent units. An action of the gauge group having this property is called k+1-fold transitive. We construct examples of E_0-semigroups of type II and index 1 which are not 2-fold transitive. These new examples also illustrate that an E_0-semigroup of type II_k need not be a tensor product of an E_0-semigroup of type II_0 and another of type I_k.
We show that the limit laws of random matrices, whose entries are conditionally independent operator valued random variables having equal second moments proportional to the size of the matrices, are operator valued semicircular laws. Furthermore, we prove an operator valued analogue of Voiculescus asymptotic freeness theorem. By replacing conditional independence with Boolean independence, we show that the limit laws of the random matrices are Operator-valued Bernoulli laws.
Let $S subset R$ be an arbitrary subset of a unique factorization domain $R$ and $K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $mathrm{Int}(S,R)= { f in mathbb{K}[x]: f(a) in R forall a in S }.$ This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call $d$-sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in $mathrm{Int}(S,R).$ In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in $mathrm{Int}(S,R).$ At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.
We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{ast} $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.
We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $mathcal{A}$. We define an $mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $mathcal{A}$-valued law $mu_t$, such that the mean and variance of $mu_t$ are continuous functions of $t$. We relate $mathcal{A}$-valued Loewner chains to processes with $mathcal{A}$-valued free or monotone independent independent increments just as was done in the scalar case by Bauer (Lowners equation from a non-commutative probability perspective, J. Theoretical Prob., 2004) and Schei{ss}inger (The Chordal Loewner Equation and Monotone Probability Theory, Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). We show that the Loewner equation $partial_t F_t(z) = DF_t(z)[V_t(z)]$, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains $F_t$ and vector fields $V_t(z)$ of the form $V_t(z) = -G_{ u_t}(z)$ where $ u_t$ is a generalized $mathcal{A}$-valued law. Based on the Loewner equation, we derive a combinatorial expression for the moments of $mu_t$ in terms of $ u_t$. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws $mu_t$. Finally, we prove a version of the monotone central limit theorem which describes the behavior of $F_t$ as $t to +infty$ when $ u_t$ has uniformly bounded support.
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