MINIMAL E_{0}-SEMIGROUPS

William Arveson

Department of Mathematics University of California Berkeley CA 94720, USA

6 December 1995

Abstract. It is known that every semigroup of normal completely positive maps of a von Neumann can be “dilated” in a particular way to an E_{0}-semigroup acting on a larger von Neumann algebra. The E_{0}-semigroup is not uniquely determined by the completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of minimality are met. Minimality is a subtle property, and it is often not obvious if it is satisfied for specific examples even in the simplest case where the von Neumann algebra is B(H).

In this paper we clarify these issues by giving a new characterization of minimality in terms of projective cocycles and their limits. Our results are valid for semigroups of endomorphisms acting on arbitrary von Neumann algebras with separable predual.

1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05. Key words and phrases. von Neumann algebras, automorphism groups, E_{0}-semigroups, mini-

mal dilations. This research was supported by NSF grants DMS92-43893 and DMS95-00291

Typeset by AMS-TEX

1. Dilations and compressions. Let M be a von Neumann algebra with separable predual. When it is convenient to do so, we will consider that M is a von Neumann subalgebra of the algebra B(H) of all bounded operators on a separable Hilbert space H which contains the identity operator 1. The separability of H will be essential for some of the results below.

By an E_{0}-semigroup acting on M we mean a family of normal ∗-endomorphisms α = {α_{t} : t ≥ 0} of M satisfying α_{t}(1) = 1 for every t ≥ 0, which obeys the semigroup property α_{s+t} = α_{s}α_{t}, and which is continuous in the sense that for every a ∈ M and every pair of vectors ξ, η ∈ H, the function t ∈ [0, ∞) ↦ ⟨α_{t}(a)ξ, η⟩ is continuous. We will also consider semigroups ϕ = {ϕ_{t} : t ≥ 0} of normal completely positive maps acting on certain von Neumann subalgebras N ⊆ M. These subalgebras will normally not contain the unit of M; but we will require that such semigroups ϕ be unital in the sense that ϕ_{t}(1_{N}) = 1_{N}, t ≥ 0, and that they should satisfy the natural continuity property cited above. We will refer to such a semigroup ϕ_{t} : N → N simply as a completely positive semigroup.

α can be compressed to certain hereditary subalgebras of M so as to give a completely positive semigroup as follows. Let M_{0} = pMp be a hereditary von Neumann subalgebra of M with unit p. The natural projection

E_{0} : M → M_{0}

of M onto M_{0} is defined by E_{0}(a) = pap. E_{0} carries the unit of M to that of M_{0}, and we have

E_{0}(axb) = aE_{0}(x)b, a, b ∈ M_{0}, x ∈ M.

Fix t ≥ 0. It is an elementary exercise to show that in order for there to exist a linear map ϕ_{t} : M_{0} → M_{0} satisfying E_{0} ∘ α_{t} = ϕ_{t} ∘ E_{0} it is necessary and sufficient that α_{t}(p) ≥ p. Thus we will be concerned with hereditary subalgebras M_{0} = pMp for which the projection p is increasing in the sense that

(1.1)  α_{t}(p) ≥ p,  t ≥ 0.

In this case one can define a family ϕ = {ϕ_{t} : t ≥ 0} of completely positive maps of M_{0} by compressing each map α_{t} to M_{0},

(1.2)  ϕ_{t}(a) = pα_{t}(a)p,  a ∈ M_{0}, t ≥ 0.

Since ϕ_{t}(p) = p, we may consider ϕ_{t} to be a unital map of M_{0}. Note too that the family ϕ has the semigroup property ϕ_{s+t} = ϕ_{s}ϕ_{t} for s, t ≥ 0. Indeed, since pα_{s}(p) = α_{s}(p)p = p by (1.1) we have

ϕ_{s}ϕ_{t}(a) = pα_{s}(pα_{t}(a)p)p = pα_{s}(p)α_{s+t}(a)α_{s}(p)p = ϕ_{s+t}(a),

for every a ∈ M_{0}. Thus ϕ is a completely positive semigroup acting on M_{0}. Throughout this paper we will be concerned with properties of completely positive semigroups which can be obtained from a fixed E_{0}-semigroup in this particular way.

Definition 1.3. Let p be a projection in M satisfying (1.1), and let pMp be the corresponding hereditary subalgebra. The completely positive semigroup ϕ = {ϕ_{t} : t ≥ 0} defined on pMp by (1.2) is called a compression of α, and α is called a dilation of ϕ.

We emphasize that the notion of a compression of α to a subalgebra has meaning only when the subalgebra is (1) a hereditary subalgebra pMp of M and (2) p is a projection satisfying (1.1). It is possible to conjure up other definitions of “compressions” of α and “dilations” of ϕ. For example, one might imagine using a conditional expectation from M onto a unital subalgebra N of M to attempt to define a semigroup of completely positive maps of N under appropriate conditions. While there has been some limited success for such endeavors [8],[9], a recent theorem of B. V. R. Bhat [5] has led us to the conclusion that the proper context for this kind of dilation theory is the context of Definition 1.3.

More precisely, Bhat has shown that for every completely positive semigroup ϕ = {ϕ_{t} : t ≥ 0} acting on a von Neumann algebra M_{0}, there is a larger von Neumann algebra M containing M_{0} as a hereditary subalgebra M_{0} = pMp and an E_{0}-semigroup α = {α_{t} : t ≥ 0} acting on M which satisfies α_{t}(p) ≥ p for every t and is such that ϕ is obtained from α by compression. Thus we have a dilation theory which resembles the more familiar dilation theory for operator semigroups which asserts that every semigroup of contraction operators on a Hilbert space can be dilated to a semigroup of isometries on a larger Hilbert space.

In the case of operator semigroups there is a simple notion of minimal isometric dilation, and two minimal isometric dilations of the same contraction semigroup are naturally unitarily equivalent. There is an analogous notion of minimality in the current setting and there is an analogous uniqueness result for the minimal E_{0}-semigroup dilation of a completely positive semigroup [5], [6]. However, these considerations for completely positive semigroups and their E_{0}-semigroup dilations are much more subtle than their counterparts in operator theory. To illustrate the level of subtlety, recall that a semigroup of contraction operators can actually be dilated further to a semigroup of unitary operators. That is because any semigroup of isometries acting on a Hilbert space is the restriction to an invariant subspace of a semigroup of unitary operators acting on a larger Hilbert space. The unitary semigroup is unique (up to a natural unitary equivalence) if the invariant subspace is “minimal”. Nothing like that is true in this setting. Indeed, an E_{0}-semigroup acting on a type I factor M does not have a natural extension to a semigroup of automorphisms of a larger type I factor which contains M as a unital subfactor. It is true that every E_{0}-semigroup can be so extended, but the construction of the extension is quite indirect and there is apparently no uniqueness of such extensions (see [2], or see [4] for a different proof). Finally, the notion of minimality introduced in [5], [6] is geared to quantum probability theory and does not lend itself readily to the natural questions that arise in the theory of E_{0}-semigroups.

The purpose of this paper is to clarify the issue of minimality for E_{0}-semigroups acting on arbitrary von Neumann algebras, and to give a new characterization of minimality in terms of the natural objects of operator algebras.

Let p be an increasing projection in M and let ϕ = {ϕ_{t} : t ≥ 0} be the compression of α to pMp. Notice that ϕ is itself an E_{0}-semigroup (acting on pMp) iff for every t ≥ 0 we have

(1.4)  ϕ_{t}(ab) = ϕ_{t}(a)ϕ_{t}(b),  a, b ∈ pMp.

Definition 1.5. Let α be an E_{0}-semigroup acting on a von Neumann algebra M. A compression ϕ of α to a hereditary subalgebra which satisfies property (1.4) is called multiplicative.

Suppose that ϕ is a compression of an E_{0}-semigroup α to a hereditary subalgebra pMp of M, and that q is an increasing projection such that q ≥ p and the compression of α to qMq is multiplicative. Then may consider that the compression of α to the intermediate subalgebra qMq is itself an E_{0}-semigroup which has ϕ as a compression.

Definition 1.6. Let α be an E_{0}-semigroup acting on a von Neumann algebra M, let p be an increasing projection in M, and let ϕ be the completely positive semigroup on pMp obtained by compression. α is said to be minimal over ϕ if the only increasing projection q ∈ M which satisfies q ≥ p, and is such that the comression of α to qMq is multiplicative, is the projection q = 1.

In order to discuss minimality further, one needs to know more about increasing projections which define multiplicative compressions. Here is the simplest class of examples. Let p be a projection of M which is fixed under α in the sense that

α_{t}(p) = p, t ≥ 0.

In this case it is clear that the compression of α to the hereditary subalgebra pMp is multiplicative. However, such projections p do not exhaust the possibilities as the following observation shows.

Proposition 1.7. Let p be an increasing projection in M and let ϕ be the compression of α to pMp. Then ϕ is multiplicative iff p commutes with α_{t}(pMp) for every t ≥ 0.

proof. Let ϕ_{t}(a) = pα_{t}(a)p, a ∈ pMp. If p commutes with α_{t}(pMp) then it is clear that (1.4) is satisfied. Conversely, if (1.4) is satisfied then for every a ∈ pMp we have

pα_{t}(a)^{∗}(1 − p)α_{t}(a)p = pα_{t}(a^{∗}a)p − pα_{t}(a^{∗})pα_{t}(a)p = ϕ_{t}(a^{∗}a) − ϕ_{t}(a^{∗})ϕ_{t}(a) = 0,

and hence (1 − p)α_{t}(a)p = 0. Thus the range of p is invariant under the self-adjoint family of operators α_{t}(pMp), hence p ∈ α_{t}(pMp)′ □

While the criterion of Proposition 1.7 is quite specific, it does not provide useful information for finding the multiplicative compressions of α. Notice for example that the family of von Neumann algebras α_{t}(pMp) appearing there neither increases nor decreases with t, because while the projections α_{t}(p) increase with t the von Neumann algebras α_{t}(M) decrease with t. In particular, Proposition 1.7 provides no insight into the order structure of the family of multiplicative compressions of α. In section 3 we will prove the following two results concerning minimality.

Theorem A. Let α be an E_{0}-semigroup acting on a von Neumann algebra M with separable predual, let p be an increasing projection in M, and let ϕ be the compression of α to the hereditary subalgebra pMp.

There is an increasing projection p_{+} ≥ p which defines a multiplicative compression of α such that if q is any other increasing projection satisfying q ≥ p for which the compression of α to qMq is multiplicative, then q ≥ p_{+}.

The compression of α to p_{+}Mp_{+} defines an E_{0}-semigroup dilation of ϕ which is minimal over ϕ.

Remark. Consider M to be a subalgebra of B(H) containing the identity operator. The following result identifies the subspace p_{+}H in concrete terms and gives an algebraic criterion for minimality in the important case where M is a factor.

Theorem B. Let α be an E_{0}-semigroup acting on a factor M ⊆ B(H), H being a separable Hilbert space. Let p ∈ M be an increasing projection and let ϕ be the compression of α to pMp. The following are equivalent.

(1) α is minimal over ϕ.

(2) H is spanned by

{α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n})ξ : a_{1}, …, a_{n} ∈ pMp, t_{k} ≥ 0, n ≥ 1, ξ ∈ pH}.

(3) M is generated as a von Neumann algebra by the set of operators

{α_{t}(a) : a ∈ pMp, t ≥ 0}.

2. Projective cocycles. Theorems A and B will be proved in section 3. They depend on properties of certain families of projections satisfying a cocycle equation.

Definition 2.1. Let α be an E_{0}-semigroup acting on M. A projective cocycle is a family of nonzero projections {p_{t} : t > 0} in M satisfying the following two conditions

(2.1.1)  p_{t} ∈ α_{t}(M)′
(2.1.2)  p_{s+t} = p_{s}α_{s}(p_{t}),  s, t > 0.

Notice that we have imposed no regularity condition on the behavior of p_{t} with respect to t. This will give us the flexibility we need for constructing examples. Nevertheless, projective cocycles are continuous:

Proposition 2.2. Let p = {p_{t} : t > 0} be a projective cocycle. Then p_{t} is a strongly continuous function of t ∈ (0, ∞), and p_{t} tends strongly to 1 as t → 0+.

proof. The family of projections p = {p_{t} : t > 0} determines a family of normal self-adjoint maps β = {β_{t} : t > 0} of M into itself by way of

β_{t}(a) = p_{t}α_{t}(a), a ∈ M, t > 0.

Because of (2.1.1) each β_{t} is an endomorphism of M, and (2.1.2) implies that β has the semigroup property β_{s+t} = β_{s}β_{t}, s, t > 0. We have β_{t}(1) = p_{t} for every t > 0.

Notice next that for fixed ξ, η ∈ H, the function t ∈ (0, ∞) ↦ ⟨p_{t}ξ, η⟩ is Borelmeasurable. Indeed, because of (2.1.1) and (2.1.2),

p_{s+t} = p_{s}α_{s}(p_{t}) ≤ p_{s}

for every s, t > 0 and hence p_{t} is decreasing in t. It follows that for every ξ ∈ H, the function

t ∈ (0, ∞) ↦ ⟨p_{t}ξ, ξ⟩ ∈ ℝ^{+}

is decreasing, therefore continuous except on a countable set, therefore measurable. The assertion about measurability of t ↦ ⟨p_{t}ξ, η⟩ follows by polarization.

This implies that β_{t} is weakly measurable in t in the sense that for every ξ, η ∈ H and every a ∈ M, the function ⟨β_{t}(a)ξ, η⟩ is Borel measurable. It follows that for every normal linear functional ρ ∈ M_{∗},

t ∈ (0, ∞) ↦ ρ(β_{t}(a)) ∈ ℂ

is measurable. Since M_{∗} is separable we can apply Proposition 2.5 of [1] to conclude that for every ρ ∈ M_{∗} we have

lim_{t→0+} ∥ρ ∘ β_{t} − ρ∥ = 0,

and

lim_{t→t_{0}} ∥ρ ∘ β_{t} − ρ ∘ β_{t_{0}}∥ = 0 for every t_{0} > 0.

In particular, taking ρ(a) = ⟨aξ, η⟩ for fixed ξ, η ∈ H we conclude that the function ⟨p_{t}ξ, η⟩ = ⟨β_{t}(1)ξ, η⟩ is continuous in t on the interval (0, ∞) and that it tends to ⟨ξ, η⟩ = ⟨1ξ, η⟩ as t → 0+.

The strong continuity of {p_{t}} asserted in Prop. (2.2) follows because the strong and weak operator topologies coincide on the set of projections □

Thus, one may always assume that a projective cocycle p = {p_{t} : t ≥ 0} is defined on the entire nonnegative real axis, and satisfies the following two conditions in addition to the two properties of Definition 2.1:

(2.1.3)  p_{0} = 1,
(2.1.4)  t ∈ [0, ∞) ↦ p_{t} is strongly continuous.

Remarks. Such cocycles have arisen in Powers’ recent work [12] on semigroups of endomorphisms of type I factors M. Given such a cocycle p = {p_{t} : t ≥ 0}, one can form the associated semigroup of (nonunital) endomorphisms of M

β_{t}(a) = p_{t}α_{t}(a).

Powers calls such a semigroup a compression of α, and he shows that the set of all compressions of α is a conditionally complete lattice with respect to its natural ordering. The compressions of particular interest in [12] are the minimal ones, i.e., those of the form

β_{t}(a) = u_{t}au_{t}^{∗}

where {u_{t} : t ≥ 0} is a semigroup of isometries satisfying

α_{t}(a)u_{t} = u_{t}a, a ∈ M, t ≥ 0.

Notice that in this case the projective cocycle p is related to the semigroup u by p_{t} = u_{t}u_{t}^{∗}, t ≥ 0.

We will not use Powers’ terminology in this paper because we are concerned with dilation theory, and in dilation theory the term compression carries a somewhat broader meaning. Moreover, our need for projective cocycles has grown from considerations that are quite different from those of [12], and it will be more convenient for us to deal directly with the cocycles rather than with their associated semigroups.

Definition 2.3. Let p = {p_{t} : t > 0} and q = {p_{t} : t > 0} be two projective cocycles. We write p ≤ q if p_{t} ≤ q_{t} for every t > 0.

The following proposition gives a general procedure for constructing projective cocycles in arbitrary von Neumann algebras from families of projections having somewhat less structure.

Proposition 2.4. Let α be an E_{0}-semigroup acting on a von Neumann algebra M and let {f_{t} : t > 0} be a family of nonzero projections in M satisfying

(2.4.1)  f_{t} ∈ α_{t}(M)′
(2.4.2)  f_{s+t} ≤ f_{s}α_{s}(f_{t}),  s, t > 0.

Fix t > 0 and consider the set of all finite partitions

P = {0 < t_{1} < t_{2} < ⋯ < t_{n} = t}

of the interval [0, t] as an increasing directed set in the usual sense. For such a partition P, define an operator f_{P} by

(2.4.3)  f_{P} = f_{t_{1}}α_{t_{1}}(f_{t_{2}−t_{1}})α_{t_{2}}(f_{t_{3}−t_{2}}) … α_{t_{n−1}}(f_{t_{n}−t_{n−1}}).

f_{P} is a projection and P_{1} ⊆ P_{2} ⟹ f_{P_{1}} ≤ f_{P_{2}}. Thus we can define a projection p_{t} by

p_{t} = sup_{P} f_{P} = lim_{P} f_{P}.

The family p = {p_{t} : t > 0} is a projective cocycle, and is the smallest projective cocycle p such that f_{t} ≤ p_{t} for every t > 0.

proof. Let s, t > 0 and let a and b be operators in M such that a commutes with α_{s}(M) and b commutes with α_{t}(M). Then a commutes with α_{s}(b) and note that the product aα_{s}(b) commutes with α_{s+t}(M). Indeed, for arbitrary c ∈ M we have

aα_{s}(b)α_{s+t}(c) = aα_{s}(bα_{t}(c)) = α_{s}(α_{t}(c)b)a = α_{s+t}(c)α_{s}(b)a = α_{s+t}(c)aα_{s}(b).

Now fix t > 0. It follows from the preceding remarks that the operator f_{P} of (2.4.3) belongs to M ∩ α_{t}(M)′; moreover, the n factors of f_{P} on the right side of (2.4.3) are mutually commuting projections. Thus f_{P} is a projection in M ∩α_{t}(M)′.

To show that f_{P} increases with P it is enough to show that if a given partition P = {0 < t_{1} < ⋯ < t_{n} = t} is refined by adjoining to it a single point τ, then f_{P} increases. In turn, that reduces to the following assertion. For k = 1, 2, …, n and t_{k−1} < τ < t_{k} (where t_{0} is taken as 0),

f_{t_{k}−t_{k−1}} ≤ f_{τ−t_{k−1}}α_{τ−t_{k−1}}(f_{t_{k}−τ}).

The latter is immediate from the hypothesis (2.4.2).

Thus the net f_{P} increases with P and we can define a projection p_{t} ∈ M ∩α_{t}(M)′ as asserted. The cocycle property (2.1.2) follows immediately from the definition of the family {p_{t} : t > 0}.

We obviously have f_{t} ≤ p_{t} for every t > 0. Finally, suppose q = {q_{t} : t > 0} is another projective cocycle satisfying f_{t} ≤ q_{t} for every t > 0. Fix t > 0 and let P = {0 < t_{1} < ⋯ < t_{n} = t} be a partition of the interval [0, t]. Then for every k = 1, 2, …, n we have f_{t_{k}−t_{k−1}} ≤ q_{t_{k}−t_{k−1}}, and hence f_{P} ≤ q_{P}. On the other hand, the cocycle property of q implies that q_{P} = q_{t}. Hence f_{P} ≤ q_{t} and we deduce the desired inequality

p_{t} = sup_{P} f_{P} ≤ q_{t}.

□

The following result will be important for section 3. It implies that certain projections in M naturally give rise to projective cocycles.

Corollary 2.5. Let e be a nonzero projection in M satisfying α_{t}(e) ≥ e for every t ≥ 0. For each t > 0 let f_{t} be the smallest projection in M ∩ α_{t}(M)′ which dominates e, i.e.,

f_{t} = [α_{t}(a)eξ : a ∈ M, ξ ∈ H].

Then f_{s+t} ≤ f_{s}α_{s}(f_{t}) for every s, t > 0. The projective cocycle p = {p_{t} : t > 0} of Proposition 2.4 is the smallest projective cocycle satisfying p_{t} ≥ e for every t > 0.

proof. It is obvious that f_{t} commutes with α_{t}(M), and the double commutant theorem implies that f_{t} ∈ M. Hence f_{t} ∈ M ∩ α_{t}(M)′.

To see that f_{s+t} ≤ f_{s}α_{s}(f_{t}), fix s, t > 0. By the argument at the beginning of the proof of proposition 2.4, f_{s}α_{s}(f_{t}) is a projection in M ∩ α_{s+t}(M)′. We claim that e ≤ f_{s}α_{s}(f_{t}). Indeed, e ≤ f_{s} follows from the definition of f_{s}, and since e ≤ f_{t} implies α_{s}(e) ≤ α_{s}(f_{t}) we have e ≤ α_{s}(e) ≤ α_{s}(f_{t}). Hence e ≤ f_{s}α_{s}(f_{t}). Since f_{s+t} is the smallest projection in M ∩ α_{s+t}(M)′ which dominates e we have the asserted inequality f_{s+t} ≤ f_{s}α_{s}(f_{t}) □

3. Minimality.

Throughout the section, α will denote an E_{0}-semigroup acting on a von Neumann algebra M ⊆ B(H) and p will denote a projection in M satisfying

(3.1)  α_{t}(p) ≥ p,  t ≥ 0.

M_{0} will denote the hereditary subalgebra pMp. We will be concerned with the (perhaps nonunital) von Neumann algebra M_{+} generated by M_{0} and its translates under α,

M_{+} = \overline{span}\{α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n}) : a_{k} ∈ M_{0}, t_{k} ≥ 0, n = 1, 2, … \},

the bar denoting closure in the weak operator topology. It is obvious that α_{t}(M_{+}) ⊆ M_{+} for every t ≥ 0, and the unit of M_{+} is the projection

(3.3)  p_{∞} = lim_{t→∞} α_{t}(p).

There is a smaller projection that is of greater importance, namely

(3.4) p_{+} = [M_{+}pH] = [α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n})ξ : a_{k} ∈ M_{0}, t_{k} ≥ 0, ξ ∈ pH].

Remarks. p_{+} is the unit of the two-sided ideal \overline{span}M_{+}pM_{+} of M_{+} generated by p. Thus, p_{+} belongs to the center of M_{+}, and in fact is the smallest central projection c in M_{+} satisfying p ≤ c.

We will eventually show that p_{+} is an increasing projection which defines a multiplicative compression of α. Neither of these assertions is apparent from (3.4). We deduce these properties from the following result which gives a “formula” for p_{+}.

Theorem 3.5. Let p be a projection in M which satisfies (3.1), let p_{∞} and p_{+} be defined as in (3.3) and (3.4) respectively. Let q = {q_{t} : t > 0} be the smallest projective cocycle satisfying q_{t} ≥ p for every t ≥ 0 as in Corollary 2.5, and set

q_{∞} = lim_{t→∞} q_{t}.

Then p_{∞} belongs to the tail von Neumann algebra M_{∞} = ∩_{t≥0}α_{t}(M), q_{∞} belongs to its relative commutant in M, and we have a factorization

p_{+} = p_{∞}q_{∞}.

Remarks. Recall that since q is a projective cocycle, q_{t} must be a decreasing function of t and hence the strong limit q_{∞} = lim_{t→∞} q_{t} exists.

proof. Notice first that since α_{t}(p) ∈ α_{t}(M) and since the von Neumann algebras α_{t}(M) decrease as t increases, it follows that p_{∞} = lim_{t→∞} p_{t} ∈ M_{∞}. Since q_{t} belongs to M ∩α_{t}(M)′ ⊆ M ∩M′_{∞} for all t we see that q_{∞} = lim_{t→∞} q_{t} ∈ M ∩M′_{∞}. In particular, the projections p_{∞} and q_{∞} must commute.

We show first that p_{+} ≤ p_{∞}q_{∞}. Since p_{+} ≤ p_{∞} is obvious, it suffices to show that p_{+}H ≤ q_{∞}H. Considering the definition of p_{+} and the fact that pH ≤ q_{∞}H, it suffices to show that the subspace q_{∞}H is invariant under any operator in any one of the von Neumann algebras α_{t}(M_{0}), t > 0, i.e., that q_{∞} commutes with ∪_{t>0}α_{t}(M_{0}). For that, fix t > 0. If we pass s to ∞ in the cocycle formula

q_{t}α_{t}(q_{s}) = q_{t+s}

and use normality of α_{t} we obtain

(3.6)  q_{t}α_{t}(q_{∞}) = q_{∞}.

It follows that for any a ∈ M_{0} we have

(3.7)  α_{t}(a)q_{∞} = α_{t}(a)q_{t}α_{t}(q_{∞}) = q_{t}α_{t}(a)α_{t}(q_{∞}) = q_{t}α_{t}(aq_{∞}),

where we have used q_{t} ∈ α_{t}(M)′. Now since p ≤ q_{∞} and since a = pap we have aq_{∞} = a = q_{∞}a. Thus we can replace the right side of (3.7) with

q_{t}α_{t}(q_{∞}a) = q_{t}α_{t}(q_{∞})α_{t}(a) = q_{∞}α_{t}(a).

Thus α_{t}(a) commutes with q_{∞} as required.

It remains to show that p_{∞}q_{∞} ≤ p_{+}. For that, it suffices to show that for every t > 0 we have

(3.8)  q_{t}α_{t}(p_{+}) ≤ p_{+}.

Indeed, assuming that (3.8) has been established we deduce q_{t}α_{t}(p) ≤ p_{+} for every t (because p ≤ p_{+}); noting that q_{t} ↓ q_{∞} and α_{t}(p) ↑ p_{∞} as t increases to +∞, we may take the strong limit on t in the previous formula to obtain the desired inequality q_{∞}p_{∞} ≤ p_{+}.

In order to prove (3.8), we require Lemma 3.9. For each t > 0 let f_{t} be the projection onto the subspace

[α_{t}(a)pξ : a ∈ M, ξ ∈ H].

Then f_{t}α_{t}(p_{+}) ≤ p_{+}.

proof. We have already pointed out in the remarks following (3.4) that p_{+} is the unit of the ideal \overline{span}M_{+}pM_{+} in M_{+}. Thus it suffices to show that

f_{t}α_{t}(M_{+}pM_{+})H ⊆ p_{+}H.

Since f_{t} commutes with α_{t}(M) the left side is contained in

α_{t}(M_{+}pM_{+})f_{t}H ⊆ [α_{t}(M_{+}pM_{+})α_{t}(M)pH] ⊆ [α_{t}(M_{+}pM_{+}M)pH].

Noting that p = α_{t}(p)p the latter is

[α_{t}(M_{+}pM_{+}Mp)pH] ⊆ [α_{t}(M_{+}pMp)pH] ⊆ [α_{t}(M_{+})pH]
⊆ [M_{+}pH] = p_{+}H,

as asserted □

For every t > 0 we define a normal linear mapping β_{t} : M → M as follows,

β_{t}(a) = f_{t}α_{t}(a).

Since f_{t} commutes with α_{t}(M), β_{t} is an endomorphism of the ∗-algebra structure of M for which β_{t}(1) = f_{t}, but it is not a semigroup because {f_{t} : t > 0} does not satisfy the cocycle condition (1.1.2). However, because of Lemma 3.9 we have

β_{t}(p_{+}) ≤ p_{+}, t > 0.

Now fix t > 0 and let P = {0 = t_{0} < t_{1} < ⋯ < t_{n} = t} be a partition of the interval [0, t]. By iterating the preceding formula we find that

(3.10)  β_{t_{1}}β_{t_{2}−t_{1}}β_{t_{3}−t_{2}} … β_{t_{n}−t_{n−1}}(p_{+}) ≤ p_{+}.

In the notation of Proposition 1.4, the left side of (3.10) is

f_{t_{1}}α_{t_{1}}(f_{t_{2}−t_{1}})α_{t_{2}}(f_{t_{3}−t_{2}}) … α_{t_{n−1}}(f_{t_{n}−t_{n−1}})α_{t}(p_{+}) = f_{P}α_{t}(p_{+}).

Using 1.4 and 1.5, we make take the limit on P in (3.10) obtain the required inequality (3.8)

q_{t}α_{t}(p_{+}) = lim_{P} f_{P}α_{t}(p_{+}) ≤ p_{+},

completing the proof of Theorem 3.5 □

We can now deduce the following result, which paraphrases Theorem A from section 1.

Theorem A. Let p be an increasing projection for α and let p_{+} ≥ p be the projection defined by (3.4). p_{+} is an increasing projection with the property that the compression of α to p_{+}Mp_{+} is multiplicative.

If r is another increasing projection in M such that r ≥ p and the compression of α to rMr is multiplicative, then r ≥ p_{+}.

proof. Since p_{∞} = lim_{t→∞} α_{t}(p) is clearly fixed under the action of α_{t} and since (3.6) implies that α_{t}(q_{∞}) ≥ q_{∞}, we find that α_{t}(p_{+}) = α_{t}(p_{∞})α_{t}(q_{∞}) ≥ p_{∞}q_{∞} = p_{+}.

To show that the compression of α to p_{+}Mp_{+} is multiplicative, it suffices to show that p_{+} commutes with α_{t}(p_{+}Mp_{+}) for every t > 0 (Propostion 1.7). For that, it is enough to show that for every a = a^{∗} ∈ p_{+}Mp_{+} we have

(3.11)  p_{+}α_{t}(a) = q_{t}α_{t}(a).

Indeed, by taking adjoints in (3.11) we find that α_{t}(a)q_{t} = α_{t}(a)p_{+}, and since q is a projective cocycle q_{t} must commute with α_{t}(M). Thus

p_{+}α_{t}(a) = q_{t}α_{t}(a) = α_{t}(a)q_{t} = α_{t}(a)p_{+},

and thus p_{+} ∈ α_{t}(p_{+}Mp_{+})′.

To prove (3.11), we write p_{+} = p_{∞}q_{∞} = q_{∞}p_{∞} and use α_{t}(p_{∞}) = p_{∞} to obtain

(3.12)  p_{+}α_{t}(a) = q_{∞}p_{∞}α(t(a) = q_{∞}α_{t}(p_{∞}a) = q_{∞}α_{t}(a),

because p_{∞}a = a for every operator a in p_{+}Mp_{+} ⊆ p_{∞}Mp_{∞}. Using (3.6) on the last term of (3.12) we have

q_{∞}α_{t}(a) = q_{t}α_{t}(p_{∞})α_{t}(a) = q_{t}α_{t}(q_{∞}a) = q_{t}α_{t}(a),

since q_{∞}a = a for every a ∈ p_{+}Mp_{+} ⊆ q_{∞}Mq_{∞}. Formula (3.11) follows.

Finally, let r ≥ p be another increasing projection with the property that the compression of α to rMr is multiplicative. We have to show that p_{+}H ⊆ rH. Because of formula (3.4) for p_{+}H, together with the fact that pH ⊆ rH, it is enough to show that rH is invariant under any operator of the form α_{t}(a) with a ∈ pMp and t > 0. But for each t > 0, Proposition 1.7 implies that r commutes with the set of operators α_{t}(rMr), and therefore since p ≤ r we have

α_{t}(pMp)rH ⊆ α_{t}(rMr)rH ⊆ rH,

as required □

As another consequence of Theorem 3.5 we have the following characterization of minimality in terms of projective cocycles.

Corollary 3.13. Let p be an increasing projection and let ϕ be the compression of α to pMp. Then α is minimal over ϕ iff lim_{t→∞} α_{t}(p) = 1 and the only projective cocycle q = {q_{t} : t > 0} satisfying q_{t} ≥ p for every t > 0 is the trivial cocycle q_{t} = 1.

proof. The minimality assertion is that p_{+} = 1 and from Theorem 3.5 we have p_{+} = p_{∞}q_{∞}. Thus α is minimal iff p_{∞} = q_{∞} = 1 □ Proposition 3.14. Let M_{+} be the following von Neumann subalgebra of M

M_{+} = \overline{span}\{α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n}) : a_{1}, …, a_{n} ∈ pMp, t_{1}, …, t_{n} ≥ 0, n ≥ 1\},

and let p_{+} be the projection of (3.4). Then p_{+} is the smallest projection in the center of M_{+} which dominates p, and we have p_{+}Mp_{+} = M_{+}p_{+}.

proof. While the first assertion is very elementary, we include a proof for completeness. Clearly p_{+} ∈ M_{+}, and since p_{+}H = [M_{+}pH] is invariant under M_{+} we have p_{+} ∈ M′_{+}. Hence p_{+} belongs to the center of M_{+} and p ≤ p_{+}. If c is another central projection in M_{+} for which c ≥ p, then cH clearly contains [M_{+}pH] = p_{+}H and hence c ≥ p_{+}.

Let R be the weakly closed subspace of M generated by the set of operators

{apb : a ∈ M_{+}, b ∈ M}.

R is a right ideal in M, and the range projection of R is

[RH] = [apH : a ∈ M_{+}] = p_{+}H.

Hence R = p_{+}M. Thus p_{+}Mp_{+} is spanned by RR^{∗}, i.e.,

p_{+}Mp_{+} = \overline{span}(M_{+} · pMp · M_{+}) = \overline{span}(M_{+}pM_{+}).

The right side of the preceding formula is the two-sided ideal in M_{+} generated by p which, by the preceding paragraph, is M_{+}p_{+} □

Theorem B. Let α be an E_{0}-semigroup acting on a factor M ⊆ B(H), H being a separable Hilbert space. Let p ∈ M be an increasing projection and let ϕ be the compression of α to pMp. The following are equivalent.

(1) α is minimal over ϕ.

(2) H is spanned by

{α_{t_{1}}(a_{1})α_{t_{2}}(a_{2}) … α_{t_{n}}(a_{n})ξ : a_{1}, …, a_{n} ∈ pMp, t_{k} ≥ 0, n ≥ 1, ξ ∈ pH}.

(3) M is generated as a von Neumann algebra by the set of operators

{α_{t}(a) : a ∈ pMp, t ≥ 0}.

proof of (1) ⟹ (3). If α is minimal over ϕ then p_{+} = 1, and thus by Proposition 3.14 we find that M = p_{+}Mp_{+} = M_{+}p_{+} = M_{+}, hence (3).

proof of (3) ⟹ (2). Since M = M_{+} we have [M_{+}pH] = [MpH]. The projection on the subspace on the right is the central carrier of p, which must be 1 because M is a factor. Therefore p_{+}H = H, as asserted in (2).

proof of (2) ⟹ (1). The hypothesis is that p_{+} = 1 which, by Theorem A, implies that α is minimal over ϕ □

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