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An E_0-semigroup is called q-pure if it is a CP-flow and its set of flow subordinates is totally ordered by subordination. The range rank of a positive boundary weight map is the dimension of the range of its dual map. Let K be a separable Hilbert space. We describe all q-pure E_0-semigroups of type II_0 which arise from boundary weight maps with range rank one over K. We also prove that no q-pure E_0-semigroups of type II_0 arise from boundary weight maps with range rank two over K. In the case when K is finite-dimensional, we provide a criterion to determine if two boundary weight maps of range rank one over K give rise to cocycle conjugate q-pure E_0-semigroups.
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The gauge group is computed explicitly for a family of E_0-semigroups of type II_0 arising from the boundary weight double construction introduced earlier by Jankowski. This family contains many E_0-semigroups which are not cocycle cocycle conjugate to any examples whose gauge groups have been computed earlier. Further results are obtained regarding the classification up to cocycle conjugacy and up to conjugacy for boundary weight doubles $(phi, u)$ in two separate cases: first in the case when $phi$ is unital, invertible and q-pure and $ u$ is any type II Powers weight, and secondly when $phi$ is a unital $q$-positive map whose range has dimension one and $ u(A) = (f, Af)$ for some function f such that $(1-e^{-x})^{1/2}f(x) in L^2(0,infty)$. All E_0-semigroups in the former case are cocycle conjugate to the one arising simply from $ u$, and any two E_0-semigroups in the latter case are cocycle conjugate if and only if they are conjugate.
We consider families of E_0-semigroups continuously parametrized by a compact Hausdorff space, which are cocycle-equivalent to a given E_0-semigroup beta. When the gauge group of $beta$ is a Lie group, we establish a correspondence between such families and principal bundles whose structure group is the gauge group of beta.
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 analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also F{o}lners type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no F{o}lner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a natural generalization of the uniform Roe algebra of a group, which we denote by $mathcal{R}_X$. We show that the following notions are then equivalent: (1) $X$ is domain measurable; (2) $X$ is not paradoxical; (3) $X$ satisfies the domain F{o}lner condition; (4) there is an algebraically amenable dense *-subalgebra of $mathcal{R}_X$; (5) $mathcal{R}_X$ has an amenable trace; (6) $mathcal{R}_X$ is not properly infinite and (7) $[0] ot=[1]$ in the $K_0$-group of $mathcal{R}_X$. We also show that any tracial state on $mathcal{R}_X$ is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of $X$. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of $C_r^*left(Xright)$ implies the amenability of $X$. The converse implication is false.
We study two classes of operator algebras associated with a unital subsemigroup $P$ of a discrete group $G$: one related to universal structures, and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of $P$ that reflect the space $mathcal{J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation $mathrm{C}^*_lambda(P)$ and we consider the boundary quotient $partial mathrm{C}^*_lambda(P)$ related to the minimal boundary space. We show that $partial mathrm{C}^*_lambda(P)$ is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of $P$; and (b) with respect to the equivariant C*-covers of the reduced nonselfadjoint semigroup algebra $mathcal{A}(P)$. If $P$ is an Ore semigroup, or if $G$ acts topologically freely on the minimal boundary space, then $partial mathrm{C}^*_lambda(P)$ coincides with the usual C*-envelope $mathrm{C}^*_{text{env}}(mathcal{A}(P))$ in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of nonzero algebraic integers, and the $ax+b$-semigroups over integral domains that are not a field. In particular, we show that $P$ is an Ore semigroup if and only if there exists a canonical $*$-isomorphism from $partial mathrm{C}^*_lambda(P)$, or from $mathrm{C}^*_{text{env}}(mathcal{A}(P))$, onto $mathrm{C}^*_lambda(G)$. If any of the above holds, then $mathcal{A}(P)$ is shown to be hyperrigid.
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