The Classification Of E₀-Semigroup

In this paper,some classifications of E₀-semigroup and there development are introduced. In noncommutative dynamical system,E₀-semigroups are very important.An E₀-semigroup is one parameter semigroup {α_t : t≥0} of the *-endmorphism on a Von. Neumann.In the 1980',R. T. Powers first studied the E₀-semigroup in [3],after that Alevras, Bhat, Power, Price and recent Muhly, Solel, Tsirelson all research it in detail,seeing ([2], [3], [4],[5]) for the detail.For E₀-semigroups, the classification of it is a fundamental problem,and many people have studied it getting a lot of developments. In this paper,we give an introduction to several classifications of E₀-semigroup .In section 1, we introduction some basic definations and properties about E₀-semigroup, the most important one is cocycle conjugacy,for it can be uesd to classify the E₀-semigroup.Then we introduce some results of its classification.In 1988 ,W. B. Arveson gave a compeletely classification of compeletely spatial E₀-semigroups up to cocycle conjugacy in [11],showing the following theory:Theorem: Every compeletely E₀-semigroup with index n is cocycle conjugate to E₀-semigroup of type I_n.(n = 1,2,…∞)In 1987, R. T. Powers constructed a spatial E₀-semigroup by way of the CAR algebra in [1],and Aleris, Alevras constructed a spatial E₀-semigroup by way of the CCR algebra in [13]. However,in 1988, R. T. Powersand D. W. Robinson in [4] proved that the previous two E₀-semigroups were isomorphic.In the next, the classifications by William Arveson,R. T. Powers,Alexis Alevras are dicussed. In [14],William Arveson described how the classificationproblem could be reduced to the problem of classifying certain sim- pler objects(Product systems)up to natural isomorphism,and discussed the role of Product systems in other dynamical issues related to the theory of E₀-semigroups.Throrem: For any E₀-semigroupα,let [ε_α] be the represention in E, then this relation is a bijective between the cocyle conjugacy class of E₀-semigroup and∑, and :[ε_α(?)β] = [ε_α] + [ε_β]In 1998,R. T. Powers introduced the definition of E₀-semigroup in standardform in[12],and showed that each spatial E₀-semigroup is cocycle conjugateto a E₀ -semigroup in standard form.In 2000,Alexis Alevras in [20] studied the standard for of an E₀-semigroup, showing that how to get all the E₀-semigroups in standard form cocycle conjugateto an given E₀-semigroup.Theorem: Letα,βbe the E₀-semigroup on B(H), B(κ)separately,andβis in standard form,cocycle conjugate toα,then there exesits the intertwing semigroup of isometris ofα,(?), such thatβis cocycle conjugate toα(?).Corollary 1: Let (?),v be the intertwing semigroup of isometries ofαseparately,thenα(?) is cocycle conjugate toα^v(?)there isα- cocycle {U_t} such that V_t = U_tS_tCorollary2: The number of standard E₀-semigroups within the cocycle conjugacy class of a given E₀-semigroupα, is equal to the number of orbits of the natural action of the group of localα- cocycle on the set of intertwing semigroups of isometries.In 1994, W. Arveson discribed an particular E₀-semigroup based on the classification of the metriable path space in [21],duced that every E₀-semigroup possessing sufficiently many decomposable operators must be cocycle conjugateto a CCR flow.This paper discussed this result in detail. Theorem: Letαbe a decomposable E₀-semigroup on B(H) which is nontrival in the sence that it cannot be extended to a group of automorphisms of B(H).Thenαis cocycle conjugate to a CCR flow.