We have seen that if ϕ: M_n(\C) \rightarrow M_n(\C) is a unital q-positive map and νis a type II Powers weight, then the boundary weight double (ϕ, ν) induces a unique (up to conjugacy) type II_0 E_0-semigroup. Let ϕ: M_n(\C) \rightarrow M_n(\C) and ψ: M_{n'}(\C) \rightarrow M_{n'}(\C) be unital rank one q-positive maps, so for some states ρ\in M_n(\C)^* and ρ' \in M_{n'}(\C)^*, we have ϕ(A)=ρ(A)I_n and ψ(D) = ρ'(D)I_{n'} for all A \in M_n(\C) and D \in M_{n'}(\C). We find that if νand ηare arbitrary type II Powers weights, then (ϕ, ν) and (ψ, η) induce non-cocycle conjugate E_0-semigroups if ρand ρ' have different eigenvalue lists. We then completely classify the q-corners and hyper maximal q-corners from ϕto ψ, obtaining the following result: If νis a type II Powers weight of the form ν(\sqrt{I - Λ(1)} B \sqrt{I - Λ(1)})=(f,Bf), then the E_0-semigroups induced by (ϕ,ν) and (ψ, ν) are cocycle conjugate if and only if n=n' and ϕand ψare conjugate.
W. Arveson has described a cocycle conjugacy class $U(α)$ of $E_0$-semigroup $α$ on B(H) which is a factor of type $\rm I$. Under some conditions on $α$, there is a $E_0$-semigroup $β\in U(α)$ being a flow of shifts in the sence of R.T. Powers. We study quasifree endomorphisms semigroups $α$ on the hyperfinite factor $M=π(A(K))''$ generated by the representations $π$ of the algebra of canonical anticommutation relations A(K) over a separable Hilbert space K. The type of M can be $\rm I$, $\rm II$ or $\rm III$ depending on $π$. The cocycle conjugacy class $U(α)$ is described in the terms of ini…
The subordinate E-semigroups of a fixed E-semigroup are in one-to-one correspondence with local projection-valued cocycles of that semigroup. For the CCR flow we characterise these cocycles in terms of their stochastic generators, that is, in terms of the coefficient driving the quantum stochastic differential equation of Hudson-Parthasarathy type that such cocycles necessarily satisfy. In addition various equivalence relations and order-type relations on E-semigroups are considered, and shown to work especially well in the case of those semigroups subordinate to the CCR flows by exploiting ou…
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 $(ϕ, ν)$ in two separate cases: first in the case when $ϕ$ is unital, invertible and q-pure and $ν$ is any type II Powers weight, and secondly when $ϕ$ is a …
We investigate $E_0-$semigroups on general factors, which are not necessarily of type I, and analyse associated invariants like product systems, super product systems etc. By tensoring $E_0-$semigroups on type I factors with $E_0-$semigroups on type II$_1$ factor, we produce several families (both countable and uncountable), consisting of mutually non-cocycle-conjugate of $E_0-$semigroups on the hyperfinite II$_\infty$ factor. Using CCR representations associated with quasi-free states, we construct for the first time, uncountable families consisting of mutually non-cocycle-conjugate $E_0-$sem…
An operator space analysis of quantum stochastic cocycles is undertaken. These are cocycles with respect to an ampliated CCR flow, adapted to the associated filtration of subspaces, or subalgebras. They form a noncommutative analogue of stochastic semigroups in the sense of Skorohod. One-to-one correspondences are established between classes of cocycle of interest and corresponding classes of one-parameter semigroups on associated matrix spaces. Each of these 'global' semigroups may be viewed as the expectation semigroup of an associated quantum stochastic cocycle on the corresponding matrix s…
We define tensor product decompositions of $E_0$-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent.
We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm{I}\sb{\infty}.$
Quantum stochastic cocycles provide a basic model for time-homogeneous Markovian evolutions in a quantum setting, and a direct counterpart in continuous time to quantum random walks, in both the Schrodinger and Heisenberg pictures. This paper is a sequel to one in which correspondences were established between classes of quantum stochastic cocycle on an operator space or C*-algebra, and classes of Schur-action `global' semigroup on associated matrix spaces over the operator space. In this paper we investigate the stochastic generation of cocycles via the generation of their corresponding globa…
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…
We introduce four new cocycle conjugacy invariants for E_0-semigroups on II_1 factors: a coupling index, a dimension for the gauge group, a super product system and a C*-semiflow. Using noncommutative Itô integrals we show that the dimension of the gauge group can be computed from the structure of the additive cocycles. We do this for the Clifford flows and even Clifford flows on the hyperfinite II_1 factor, and for the free flows on the free group factor $L(F_\infty)$. In all cases the index is 0, which implies they have trivial gauge groups. We compute the super product systems for these fam…
We show that every binary shift on the hyperfinite $II_1$ factor $R$ is cocycle conjugate to at least countably many non-conjugate binary shifts. This holds in particular for binary shifts of infinite commutant index.
We show that Bogoljubov actions of $\mathbb{R}$ on the free group factor $L(\mathbb{F}_{\infty})$ associated to sums of infinite multiplicity trivial and certain mixing representations are cocycle conjugate if and only if the underlying representations are conjugate.
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 consequence, we establish the existence of uncountably many non-cocycle conjugate E_0-semigroups of type II_0 which are prime.
We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on $\mathscr{B}(G)$ ($G$ a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of $E_0$-semigroups on $\mathscr{B}(H)$ by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stoc…
We prove that all strongly outer Z^N-actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable Z^N-actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other.
An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of $*$-endomorphisms of $B(H)$ that preserve the identity. Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat induction of a $CP$-flow over a separable Hilbert space $K$. We say an $E_0$-semigroup $α$ is $q$-pure if the $CP$-subordinates $β$ of norm one (i.e. $\Vertβ_t(I)\Vert = 1$ and $α_t-β_t$ is completely positive for all $t \geq 0$) are totally ordered in the sense that if $β$ and $γ$ are two $CP$-subordin…
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 ra…
Recently it is proved in arXiv:1906.05493v1 [math.OA] that CCR flows over convex cones are cocycle conjugate if and only if the associated isometric representations are conjugate. We provide a very short, simple and direct proof of that. Using the same idea we prove the analogous statement for CAR flows as well. Further we show that CCR flows are not cocycle conjugate to the CAR flows when the (multi-parameter) isometric representation is `proper', a condition which is satisfied by all known examples.
We introduce a new construction of $E_0$-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of $C_0$-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III $E_0$-semigroups, which can not be distinguished from $E_0$-semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III facto…
Powers has shown that each spatial E_0-semigroup can be obtained from the boundary weight map of a CP-flow acting on B(K \otimes L^2(0, \infty)) for some separable Hilbert space K. In this paper, we define boundary weight maps through boundary weight doubles (ϕ, ν), where ϕ: M_n(\C) \to M_n(\C) is a q-positive map and νis a boundary weight over L^2(0, \infty). These doubles induce CP-flows over K for 1<dim(K)<\infty which then minimally dilate to E_0-semigroups by a theorem of Bhat. Through this construction, we obtain uncountably many mutually non-cocycle conjugate E_0-semigroups for each n>1…
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 cocycle 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 flow…
From previous work, we know how to obtain type II_0 E_0-semigroups using boundary weight doubles (ϕ, ν), where ϕ: M_n(\C) \to M_n(\C) is a unital q-positive map and νis a normalized unbounded boundary weight over L^2(0, \infty). In this paper, we classify the unital q-positive maps ϕ: M_2(\C) \to M_2(\C). We find that every unital q-pure map ϕ: M_2(\C) \to M_2(\C) is either rank one or invertible. We also examine the case n=3, finding the limit maps L_ϕfor all unital q-positive maps ϕ: M_3(\C) \to M_3(\C). In conclusion, we present a cocycle conjugacy result for E_0-semigroups induced by bound…
Toeplitz CAR flows are a class of E_0-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate E_0-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.
Quantum random walks are constructed on operator spaces with the aid of matrix-space lifting, a type of ampliation intermediate between those provided by spatial and ultraweak tensor products. Using a form of Wiener-Ito decomposition, a Donsker-type theorem is proved, showing that these walks, after suitable scaling, converge in a strong sense to vacuum cocycles: these are vacuum-adapted processes which are Feller cocycles in the sense of Lindsay and Wills. This is employed to give a new proof of the existence of *-homomorphic quantum stochastic dilations for completely positive contraction se…
Let B be a sigma-unital C*-algebra. We show that every strongly continuous E_0-semigroup on the algebra of adjointable operators on a full Hilbert B-module E gives rise to a full continuous product system of correspondences over B. We show that every full continuous product system of correspondences over B arises in that way. If the product system is countably generated, then E can be chosen countable generated, and if E is countably generated, then so is the product system. We show that under these countability hypotheses there is a one-to-one correspondence between E_0-semigroup up to stable…
An E_0-semigroup acting on B(H) is called pure if the intersection of the ranges $α_t(B(H))$, $t>0$, is the algebra of scalars. We determine all pure E_0-semigroups which have a weakly continuous invariant state $ω$ and which are minimal in an appropriate sense. In such cases, for every normal state $ρ$ there is convergence to equilibrium in the trace norm $\lim_{t\to\infty}\| ρ\circα_t - ω\| = 0.$ A normal state $ω$ with this property is called an absorbing state. Such E_0-semigroups mest be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those pertu…
Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $ϕ$ and $θ$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $α$ and $β$ such that $ϕ_s \circ θ_t (P_H A P_H) = P_H α_s \circ β_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $ϕ$ is not an automorphism semigroup then $α$ is cocycle conjugate to the minimal *-endomorphic dilation of $ϕ$, and that if $ϕ$ is an automorphism semigroup then $α$ is also an automorphism semigroup. In particular, we conclude that if $ϕ$ is not an automorphism semigroup a…
We construct numerous continuous families of irreducible subfactors of the hyperfinite II$_1$ factor, which are non-isomorphic, but have all the same standard invariant. In particular, we obtain 1-parameter families of irreducible, non-isomorphic subfactors of the hyperfinite II$_1$ factor, which have Jones index 6 and have all the same standard invariant with property (T). We exploit the fact that property (T) groups have uncountably many non-cocycle conjugate cocycle actions on the hyperfinite II$_1$ factor.
We consider Z-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two Z-actions with the Rohlin property on such a C^*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer Z^2-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as…
In the 2002 AMS summer conference on ``Advances in Quantum Dynamics'' in Mount Holyoke Robert Powers proposed a sum operation for spatial E0-semigroups. Still during the conference Skeide showed that the Arveson system of that sum is the product of spatial Arveson systems. This product may but need not coincide with the tensor product of Arveson systems. The Powers sum of two spatial E0-semigroups is, therefore, up to cocycle conjugacy Skeide's product of spatial noises.
In this paper we use the description of free group factors as the von Neumann algebras of Berezin's deformation of the upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $$ {\cal L}(x)=Λ(x)-(1/…
