In this paper we study modular extendability and equimodularity of endomorphisms and E$_0$-semigroups on factors with respect to f.n.s. weights. We show that modular extendability is a property that does not depend on the choice of weights, it is a c
ocycle conjugacy invariant and it is preserved under tensoring. We say that a modularly extendable E$_0$-semigroup is of type EI, EII or EIII if its modular extension is of type I, II or III, respectively. We prove that all types exist on properly infinite factors. We also compute the coupling index and the relative commutant index for the CAR flows and $q$-CCR flows. As an application, by considering repeated tensors of the CAR flows we show that there are infinitely many non cocycle conjugate non-extendable $E_0$-semigroups on the hyperfinite factors of types II$_1$, II$_{infty}$ and III$_lambda$, for $lambda in (0,1)$.
We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselters generalization of the Cunt
z-Pimsner C*-algebra to the context of subproduct systems. Suppose that $X$ and $Y$ are Arveson-Stinespring subproduct systems associated to two stochastic matrices over a countable set $Omega$, and let $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ be their tensor algebras. We show that every algebraic isomorphism from $mathcal{T}_+(X)$ onto $mathcal{T}_+(Y)$ is automatically bounded. Furthermore, $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are isometrically isomorphic if and only if $X$ and $Y$ are unitarily isomorphic up to a *-automorphism of $ell^infty(Omega)$. When $Omega$ is finite, we prove that $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are algebraically isomorphic if and only if there exists a similarity between $X$ and $Y$ up to a *-automorphism of $ell^infty(Omega)$. Moreover, we provide an explicit description of the Cuntz-Pimsner algebra $mathcal{O}(X)$ in the case where $Omega$ is finite and the stochastic matrix is essential.
A CP-semigroup is aligned if its set of trivially maximal subordinates is totally ordered by subordination. We prove that aligned spatial E_0-semigroups are prime: they have no non-trivial tensor product decompositions up to cocycle conjugacy. As a c
onsequence, we establish the existence of uncountably many non-cocycle conjugate E_0-semigroups of type II_0 which are prime.
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 famil
ies and principal bundles whose structure group is the gauge group of beta.
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 sp
ace. 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.
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.
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 ch
aracterized 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.
I. Raeburn and J. Taylor have constructed continuous-trace C*-algebras with a prescribed Dixmier-Douady class, which also depend on the choice of an open cover of the spectrum. We study the asymptotic behavior of these algebras with respect to certai
n refinements of the cover and appropriate extension of cocycles. This leads to the analysis of a limit groupoid G and a cocycle sigma, and the algebra C*(G, sigma) may be regarded as a generalized direct limit of the Raeburn-Taylor algebras. As a special case, all UHF C*-algebras arise from this limit construction.
