GENERATORS OF NONCOMMUTATIVE DYNAMICS

WILLIAM ARVESON

Abstract. For a fixed C^{∗}-algebra A, we consider all noncommutative dynamical systems that can be generated by A. More precisely, an Adynamical system is a triple (i, B, α) where α is a ∗-endomorphism of a C^{∗}-algebra B, and i : A ⊆ B is the inclusion of A as a C^{∗}-subalgebra with the property that B is generated by A ∪ α(A) ∪ α^{2}(A) ∪ · · ·. There is a natural hierarchy in the class of A-dynamical systems, and there is a universal one that dominates all others, denoted (i, PA, α). We establish certain properties of (i, PA, α) and give applications to some concrete issues of noncommutative dynamics.

For example, we show that every contractive completely positive linear map φ : A → A gives rise to to a unique A-dynamical system (i, B, α) that is “minimal” with respect to φ, and we show that its C^{∗}-algebra B can be embedded in the multiplier algebra of A ⊗ K.

1. Generators

The flow of time in quantum theory is represented by a one-parameter group of ∗-automorphisms {α_{t} : t ∈ ℝ} of a C^{∗}-algebra B. There is often a C^{∗}-subalgebra A ⊆ B that can be singled out from physical considerations which, together with its time translates, generates B. For example, in nonrelativistic quantum mechanics the flow of time is represented by a one-parameter group of automorphisms of B(H), and the set of all bounded continuous functions of the configuration observables at time 0 is a commutative C^{∗}-algebra A. The set of all time translates α_{t}(A) of A generates an irreducible C^{∗}-subalgebra B of B(H). In particular, for different times t_{1} ≠ t_{2}, the C^{∗}-algebras α_{t_{1}}(A) and α_{t_{2}}(A) do not commute with each other. Indeed, no nontrivial relations appear to exist between α_{t_{1}}(A) and α_{t_{2}}(A) when t_{1} ≠ t_{2}.

In this paper we look closely at this phenomenon, in a simpler but analogous setting. Let A be a C^{∗}-algebra, fixed throught. Definition 1.1. An A-dynamical system is a triple (i, B, α) consisting of a ∗- endomorphism α acting on a C^{∗}-algebra B and an injective ∗-homomorphism i : A → B, such that B is generated by i(A) ∪ α(i(A)) ∪ α^{2}(i(A)) ∪ · · ·. We lighten notation by identifying A with its image i(A) in B, thereby replacing i with the inclusion map i : A ⊆ B. Thus, an A-dynamical system is a dynamical system (B, α) that contains A as a C^{∗}-subalgebra in a specified way, with the property that B is the norm-closed linear span of finite products of the following form

1991 Mathematics Subject Classification. 46L55, 46L09.

Partially supported by NSF grants DMS-9802474 and DMS-0100487.

B = \overline{span}\{α^{n_{1}}(a_{1})α^{n_{2}}(a_{2}) ⋯ α^{n_{k}}(a_{k})\}  (1)

where n_{1}, …, n_{k} ≥ 0, a_{1}, …, a_{k} ∈ A, k = 1, 2, …

Our aim is to say something sensible about the class of all A-dynamical systems, and to obtain more detailed information about certain of its members. The opening paragraph illustrates the fact that in even the simplest cases, where A is C(X) or even a matrix algebra, the structure of individual A-dynamical systems can be very complex.

There is a natural hierarchy in the class of all A-dynamical systems, defined by (i_{1}, B_{1}, α_{1}) ≥ (i_{2}, B_{2}, α_{2}) iff there is a ∗-homomorphism θ : B_{1} → B_{2} satisfying θ ∘ α_{1} = α_{2} ∘ θ and θ(a) = a for a ∈ A. Since θ fixes A, it follows from (1) that θ must be surjective, θ(B_{1}) = B_{2}, hence (i_{2}, B_{2}, α_{2}) is a quotient of (i_{1}, B_{1}, α_{1}). Two A-dynamical systems are said to be equivalent if there is a map θ as above that is an isomorphism of C^{∗}-algebras. This will be the case iff each of the A-dynamical systems dominates the other. One may also think of the class of all A-dynamical systems as a category, whose objects are A-dynamical systems and whose maps θ are described above.

There is a largest equivalence class in this hierarchy, whose representatives are called universal A-dynamical systems. We exhibit one as follows. Consider the free product of an infinite sequence of copies of A,

PA = A ∗ A ∗ · · · .

Thus, we have a sequence of ∗-homomorhisms θ_{0}, θ_{1}, … of A into the C^{∗}- algebra PA such PA is generated by θ_{0}(A) ∪ θ_{1}(A) ∪ ⋯ and such that the following universal property is satisfied: for every sequence π_{0}, π_{1}, … of ∗-homomorphisms of A into some other C^{∗}-algebra B, there is a unique ∗-homomorphism ρ : PA → B such that π_{k} = ρ ∘ θ_{k}, k = 0, 1, . . .. Nondegenerate representations of PA correspond to sequences π̄ = (π_{0}, π_{1}, …) of representations π_{k} : A → B(H) of A on a common Hilbert space H, subject to no condition other than the triviality of their common nullspace

ξ ∈ H, π_{k}(A)ξ = {0}, k = 0, 1, ⋯ ⟹ ξ = 0.

A simple argument establishes the existence of PA by taking the direct sum of a sufficiently large set of such representation sequences π̄.

This definition does not exhibit PA in concrete terms (see §3 for that), but it does allow us to define a universal A-dynamical system. The universal property of PA implies that there is a shift endomorphism σ : PA → PA defined uniquely by σ ∘ θ_{k} = θ_{k+1}, k = 0, 1, . . .. It is quite easy to verify that θ_{0} is an injective ∗-homomorphism of A in PA, and we use this map to identify A with θ_{0}(A) ⊆ PA. Thus the triple (i, PA, σ) becomes an Adynamical system with the property that every other A-dynamical system is subordinate to it.

Before introducing α-expectations, we review some common terminology [Ped79]. Let A ⊆ B be an inclusion of C^{∗}-algebras. For any subset S of B we write [S] for the norm-closed linear span of S. The subalgebra A is said to be essential if the two-sided ideal [BAB] it generates is an essential ideal

x ∈ B, xBAB = {0} ⟹ x = 0.

It is called hereditary if for a ∈ A and b ∈ B, one has

0 ≤ b ≤ a ⟹ b ∈ A.

The hereditary subalgebra of B generated by a subalgebra A is the closed linear span [ABA] of all products axb, a, b ∈ A, x ∈ B, and in general A ⊆ [ABA]. A corner of B is a hereditary subalgebra of the particular form A = pBp where p is a projection in the multiplier algebra M(B) of B.

We also make essential use of conditional expectations E : B → A. A conditional expectation is an idempotent positive linear map with range A, satisfying E(ax) = aE(x) for a ∈ A, x ∈ B. When A = pBp is a corner of B, the map E(x) = pxp, defines a conditional expectation of B onto A. On the other hand, many of the conditional expectations encountered here do not have this simple form, even when A has a unit. Indeed, if A is subalgebra of B that is not hereditary, then there is no natural conditional expectation E : B → A. In general, conditional expectations are completely positive linear maps with ∥E∥ = 1. Definition 1.2. Let (i, B, α) be an A-dynamical system. An α-expectation is a conditional expectation E : B → A having the following two properties:

E1. Equivariance: E ∘ α = E ∘ α ∘ E.

E2. The restriction of E to the hereditary subalgebra generated by A is multiplicative, E(xy) = E(x)E(y), x, y ∈ [ABA].

Note that an arbitrary conditional expectation E : B → A gives rise to a linear map φ : A → A by way of φ(a) = E(α(a)), a ∈ A. Such a φ is a completely positive map satisfying ∥φ∥ ≤ 1. Axiom E1 makes the assertion

E ∘ α = φ ∘ E.  (2)

where φ = E ∘ α ↾_{A} is the linear map of A associated with E.

Property E2 is of course automatic if A is a hereditary subalgebra of B. It is a fundamentally noncommutative hypothesis on B. For example, if Y is a compact Hausdorff space and B = C(Y), then every unital subalgebra A ⊆ C(Y) generates C(Y) as a hereditary algebra. Thus the only linear maps E : C(Y) → A satisfying E2 are ∗-endomorphisms of C(Y). The key property of the universal A-dynamical system (i, PA, σ) follows. Theorem 1.3. For every completely positive contraction φ : A → A, there is a unique σ-expectation E : PA → A satisfying

φ(a) = E(σ(a)),  a ∈ A.  (3)

Both assertions are nontrivial. We prove uniqueness in the following section, see Theorem 2.3. Existence is taken up in §3, see Theorem 3.2.

2. moment polynomials

This theory of generators rests on properties of certain noncommutative polynomials that are defined recursively as follows. Proposition 2.1. Let A be an algebra over a field 𝔽. For every linear map φ : A → A, there is a unique sequence of multilinear mappings from A to itself, indexed by the k-tuples of nonnegative integers, k = 1, 2, …, where for a fixed k-tuple n̄ = (n_{1}, …, n_{k})

a_{1}, …, a_{k} ∈ A ↦ [n̄; a_{1}, …, a_{k}] ∈ A

is a k-linear mapping, all of which satisfy MP1. φ([n̄; a_{1}, …, a_{k}]) = [n_{1} + 1, n_{2} + 1, …, n_{k} + 1; a_{1}, …, a_{k}]. MP2. Given a k-tuple for which n_{ℓ} = 0 for some ℓ between 1 and k,

[n̄; a_{1}, …, a_{k}] = [n_{1}, …, n_{ℓ−1}; a_{1}, …, a_{ℓ−1}]a_{ℓ}[n_{ℓ+1}, …, n_{k}; a_{ℓ+1}, …, a_{k}].

Remark 2.2. The proofs of both existence and uniqueness are straightforward arguments using induction on the number k of variables, and we omit them. Note that in the second axiom MP2, we make the natural conventions when ℓ has one of the extreme values 1, k. For example, if ℓ = 1, then MP2 should be interpreted as

[0, n_{2}, …, n_{k}; a_{1}, …, a_{k}] = a_{1}[n_{2}, …, n_{k}; a_{2}, …, a_{k}].

In particular, in the linear case k = 1, MP2 makes the assertion

[0; a] = a, a ∈ A;

and after repeated applications of axiom MP1 one obtains

[n; a] = φ^{n}(a), a ∈ A, n = 0, 1, ….

One may calculate any particular moment polynomial explicitly, but the computations quickly become a tedious exercise in the arrangement of parentheses. For example,

[2, 6, 3, 4; a, b, c, d] = φ^{2}(aφ(φ^{3}(b)cφ(d))),
[6, 4, 2, 3; a, b, c, d] = φ^{2}(φ^{2}(φ^{2}(a)b)cφ(d)).

Finally, we remark that when A is a C^{∗}-algebra and φ : A → A is a linear map satisfying φ(a)^{∗} = φ(a^{∗}), a ∈ A, then its associated moment polynomials obey the following symmetry

[n_{1}, …, n_{k}; a_{1}, …, a_{k}]^{∗} = [n_{k}, …, n_{1}; a_{k}^{∗}, …, a_{1}^{∗}].  (4)

Indeed, one finds that the sequence of polynomials [[·; ·]] defined by

[[n_{1}, …, n_{k}; a_{1}, …, a_{k}]] = [n_{k}, …, n_{1}; a_{k}^{∗}, …, a_{1}^{∗}]^{∗}

also satisfies axioms MP1 and MP2, and hence must coincide with the moment polynomials of φ by the uniqueness assertion of Proposition 2.1.

These polynomials are important because they are the expectation values of certain A-dynamical systems.

Theorem 2.3. Let φ : A → A be a completely positive map on A, satisfying ∥φ∥ ≤ 1, with associated moment polynomials [n_{1}, …, n_{k}; a_{1}, …, a_{k}].

Let (i, B, α) be an A-dynamical system and let E : B → A be an αexpectation with the property E(α(a)) = φ(a), a ∈ A. Then

E(α^{n_{1}}(a_{1})α^{n_{2}}(a_{2}) ⋯ α^{n_{k}}(a_{k})) = [n_{1}, …, n_{k}; a_{1}, …, a_{k}].  (5)

for every k = 1, 2, …, n_{k} ≥ 0, a_{k} ∈ A. In particular, there is at most one α-expectation E : B → A satisfying E(α(a)) = φ(a), a ∈ A. Proof. One applies the uniqueness of moment polynomials as follows. Properties E1 and E2 of Definition 1.2 imply that the sequence of polynomials [[·; ·]] defined by

[[n_{1}, …, n_{k}; a_{1}, …, a_{k}]] = E(α^{n_{1}}(a_{1}) ⋯ α^{n_{k}}(a_{k}))

must satisfy the two axioms MP1 and MP2. Notice here that E2 implies

E(xay) = E(x)aE(y)  x, y ∈ B, a ∈ A,  (6)

since for an approximate unit e_{n} for A we can write E(xay) as follows:

lim_{n→∞} e_{n}E(xae_{n}y)e_{n} = lim_{n→∞} E(e_{n}xae_{n}ye_{n}) = lim_{n→∞} E(e_{n}xa)E(e_{n}ye_{n})

= lim_{n→∞} e_{n}E(x)ae_{n}E(y)e_{n} = E(x)aE(y).

Thus formula (5) follows from the uniqueness assertion of Proposition 2.1. The uniqueness of the α-expectation associated with φ is now apparent from formulas (5) and (1). □

3. Existence of α-expectations

In this section we show that every completely positive map φ : A → A, with ∥φ∥ ≤ 1, gives rise to a σ-expectation E : PA → A that is related to the moment polynomials of φ as in (5). This is established through a construction that exhibits PA as the enveloping C^{∗}-algebra of a Banach ∗- algebra ℓ^{1}(Σ), in such a way that the desired conditional expectation appears as a completely positive map on ℓ^{1}(Σ). The details are as follows.

Let S be the set of finite sequences n̄ = (n_{1}, n_{2}, …, n_{k}) of nonnegative integers, k = 1, 2, … which have distinct neighbors,

n_{1} ≠ n_{2}, n_{2} ≠ n_{3}, …, n_{k−1} ≠ n_{k}.

Multiplication and involution are defined in S as follows. The product of two elements m̄ = (m_{1}, …, m_{k}), n̄ = (n_{1}, …, n_{ℓ}) ∈ S is defined by conditional concatenation

m̄ · n̄ = \begin{cases}(m_{1}, …, m_{k}, n_{1}, …, n_{ℓ}),  if m_{k} ≠ n_{1}, \\ (m_{1}, …, m_{k}, n_{2}, …, n_{ℓ}),  if m_{k} = n_{1}, \end{cases}

where we make the natural conventions when n̄ = (q) is of length 1, namely m̄ · (q) = (m_{1}, …, m_{k}, q) if m_{k} ≠ q, and m̄ · (q) = m̄ if m_{k} = q. The involution in S is defined by reversing the order of components

(m_{1}, …, m_{k})^{∗} = (m_{k}, …, m_{1}).

One finds that S is an associative ∗-semigroup.

Fixing a C^{∗}-algebra A, we attach a Banach space Σ_{ν} to every k-tuple ν = (n_{1}, …, n_{k}) ∈ S as follows

Σ_{ν} = \underbrace{A ˆ⊗ ⋯ ˆ⊗A}_{k times},

the k-fold projective tensor product of copies of the Banach space A. We assemble the Σ_{ν} into a family of Banach spaces over S, p : Σ → S, by way of Σ = {(ν, ξ) : ν ∈ S, ξ ∈ E_{ν}}, p(ν, ξ) = ν.

We introduce a multiplication in Σ as follows. Fix μ = (m_{1}, …, m_{k}) and ν = (n_{1}, …, n_{ℓ}) in S and choose ξ ∈ Σ_{μ}, η ∈ Σ_{ν}. If m_{k} ≠ n_{1} then ξ · η is defined as the tensor product ξ ⊗ η ∈ Σ_{μ·ν}. If m_{k} = n_{1} then we must tensor over A and make the obvious identifications. More explicitly, in this case there is a natural map of the tensor product Σ_{μ} ⊗_{A} Σ_{ν} onto Σ_{μ·ν} by making identifications of elementary tensors as follows:

(a_{1} ⊗ ⋯ ⊗ a_{k}) ⊗_{A} (b_{1} ⊗ ⋯ ⊗ b_{ℓ}) ∼ a_{1} ⊗ ⋯ ⊗ a_{k−1} ⊗ a_{k}b_{1} ⊗ b_{2} ⊗ ⋯ ⊗ b_{ℓ}.

With this convention ξ · η is defined by

ξ · η = ξ ⊗_{A} η ∈ Σ_{μ·ν}.

This defines an associative multiplication in the family of Banach spaces Σ. There is also a natural involution in Σ, defined on each Σ_{μ}, μ = (m_{1}, …, m_{k}) as the unique antilinear isometry to Σ_{μ^{∗}} satisfying

((m_{1}, …, m_{k}), a_{1} ⊗ ⋯ ⊗ a_{k})^{∗} = ((m_{k}, …, m_{1}), a_{k}^{∗} ⊗ ⋯ ⊗ a_{1}^{∗}).

This defines an isometric antilinear mapping of the Banach space Σ_{μ} onto Σ_{μ^{∗}}, for each μ ∈ S, and thus the structure Σ becomes an involutive ∗- semigroup in which each fiber Σ_{μ} is a Banach space.

Let ℓ^{1}(Σ) be the Banach ∗-algebra of summable sections. The norm and involution are the natural ones ∥f∥ = ∑_{μ∈Σ} ∥f(μ)∥, f^{∗}(μ) = f(μ^{∗})^{∗}. Noting that Σ_{λ} · Σ_{μ} ⊆ Σ_{λ·μ}, the multiplication in ℓ^{1}(Σ) is defined by convolution

f ∗ g(ν) = ∑_{λ·μ=ν} f(λ) · g(μ),

and one easily verifies that ℓ^{1}(Σ) is a Banach ∗-algebra.

For μ = (m_{1}, …, m_{k}) ∈ S and a_{1}, …, a_{k} ∈ A we define the function

δ_{μ} · a_{1} ⊗ ⋯ ⊗ a_{k} ∈ ℓ^{1}(Σ)

to be zero except at μ, and at μ it has the value a_{1} ⊗ ⋯ ⊗ a_{k} ∈ Σ_{μ}. These elementary functions have ℓ^{1}(Σ) as their closed linear span. Finally, there is a natural sequence of ∗-homomorphisms θ_{0}, θ_{1}, ⋯: A → ℓ^{1}(Σ) defined by

θ_{k}(a) = δ_{(k)} · a, a ∈ A, k = 0, 1, …,

and these maps are related to the generating sections by

δ_{(n_{1},...,n_{k})} · a_{1} ⊗ ⋯ ⊗ a_{k} = θ_{n_{1}}(a_{1})θ_{n_{2}}(a_{2}) ⋯ θ_{n_{k}}(a_{k}).

The algebra ℓ^{1}(Σ) fails to have a unit, but it has the same representation theory as PA in the following sense. Given a sequence of representations π_{k} : A → B(H), k = 0, 1, …, fix ν = (n_{1}, …, n_{k}) ∈ S. There is a unique bounded linear operator L_{ν} : Σ_{ν} → B(H) of norm 1 that is defined by its action on elementary tensors as follows

L_{ν}(a_{1} ⊗ ⋯ ⊗ a_{k}) = π_{n_{1}}(a_{1}) ⋯ π_{n_{k}}(a_{k}).

Thus there is a bounded linear map π̃ : ℓ^{1}(Σ) → B(H) defined by

π̃(f) = ∑_{μ∈S} L_{μ}(f(μ)), f ∈ ℓ^{1}(Σ).

One finds that π̃ is a ∗-representation of ℓ^{1}(Σ) with ∥π̃∥ = 1. This representation satisfies π̃ ∘ θ_{k} = π_{k}, k = 0, 1, 2, , . . .. Conversely, every bounded ∗-representation π̃ of ℓ^{1}(Σ) on a Hilbert space H is associated with a sequence of representations π_{0}, π_{1}, … of A on H by way of π_{k} = π̃ ∘ θ_{k}.

The results of the preceding discussion are summarized as follows:

Proposition 3.1. The enveloping C^{∗}-algebra C^{∗}(ℓ^{1}(Σ)), together with the sequence of homomorphisms θ̃_{0}, θ̃_{1}, ⋯: A → C^{∗}(ℓ^{1}(Σ)) defined by the maps θ_{0}, θ_{1}, ⋯: A → ℓ^{1}(Σ), has the same universal property as the infinite free product PA = A ∗ A ∗ ⋯, and is therefore isomorphic to PA.

Notice that the natural shift endomorphism of ℓ^{1}(Σ) is defined by

σ : δ_{(n_{1},...,n_{k})} · ξ ↦ δ_{(n_{1}+1,...,n_{k}+1)} · ξ, ν = (n_{1}, …, n_{k}) ∈ Σ, ξ ∈ Σ_{ν}

and it promotes to the natural shift endomorphism of PA = C^{∗}(ℓ^{1}(Σ)). The inclusion of A in ℓ^{1}(Σ) is given by the map θ_{0}(a) = δ_{(0)}a ∈ ℓ^{1}(Σ), and it too promotes to the natural inclusion of A in PA.

Finally, we fix a contractive completely positive map φ : A → A, and consider the moment polynomials associated with it by Proposition 2.1. A straightforward argument shows that there is a unique bounded linear map E_{0} : ℓ^{1}(Σ) → A satisfying

E_{0}(δ_{(n_{1},...,n_{k})} · a_{1} ⊗ ⋯ ⊗ a_{k}) = [n_{1}, …, n_{k}; a_{1}, …, a_{k}],

for (n_{1}, …, n_{k}) ∈ S, a_{1}, …, a_{k} ∈ A, k = 1, 2, …, and ∥E_{0}∥ = ∥φ∥ ≤ 1. Using the axioms MP1 and MP2, one finds that the map E_{0} preserves the adjoint (see Equation (4)), satisfies the conditional expectation property E_{0}(af) = aE_{0}(f) for a ∈ A, f ∈ ℓ^{1}(Σ), that the restriction of E_{0} to the “hereditary” ∗-subalgebra of ℓ^{1}(Σ) spanned by θ_{0}(A)ℓ^{1}(Σ)θ_{0}(A) is multiplicative, and that it is related to φ by E_{0} ∘σ = φ∘E_{0} and E_{0}(σ(a)) = φ(a), a ∈ A. Thus, E_{0} satisfies the axioms of Definition 1.2, suitably interpreted for the Banach ∗-algebra ℓ^{1}(Σ).

In view of the basic fact that a bounded completely positive linear map of a Banach ∗-algebra to A promotes naturally to a completely positive map of its enveloping C^{∗}-algebra to A, the critical property of E_{0} reduces to:

Theorem 3.2. For every n ≥ 1, a_{1}, …, a_{n} ∈ A, and f_{1}, …, f_{n} ∈ ℓ^{1}(Σ), we have

∑_{i,j=1}^{n} a_{j}^{∗}E_{0}(f_{j}^{∗}f_{i})a_{i} ≥ 0.

Consequently, E_{0} extends uniquely through the completion map ℓ^{1}(Σ) → PA to a completely positive map E_{φ} : PA → A that becomes a σ-expectation satisfying Equation (3).

We sketch the proof of Theorem 3.2, detailing the critical steps. Using the fact that ℓ^{1}(Σ) is spanned by the generating family

G = {δ_{(n_{1},...,n_{k})} · a_{1} ⊗ ⋯ ⊗ a_{k} : (n_{1}, …, n_{k}) ∈ S, a_{1}, …, a_{k} ∈ A, k ≥ 1}

one easily reduces the proof of Theorem 3.2 to the following more concrete assertion: for any finite set of elements u_{1}, …, u_{n} in G, the n × n matrix (a_{ij}) = (E_{0}(u_{j}^{∗}u_{i})) ∈ M_{n}(A) is positive.

The latter is established by an inductive argument on the “maximum height” max(h(u_{1}), …, h(u_{n})), where the height of an element u = δ_{(n_{1},...,n_{k})}· a_{1}⊗⋯⊗a_{k} in G is defined as h(u) = max(n_{1}, …, n_{k}). The general case easily reduces to that in which A has a unit e, and in that setting the inductive step is implemented by the following. Lemma 3.3. Choose u_{1}, …, u_{n} ∈ G such that the maximum height N = max(h(u_{1}), …, h(u_{n})) is positive. For k = 1, …, n there are elements b_{k}, c_{k} ∈ A and v_{k} ∈ G such that h(v_{k}) < N and

E_{0}(u_{j}^{∗}u_{i}) = b_{j}^{∗}φ(E_{0}(v_{j}^{∗}v_{i}))b_{i} + c_{j}^{∗}(e − φ(e))c_{i},  1 ≤ i, j ≤ n. (7)

Remark 3.4. Note that if an inductive hypothesis provides positive n × n matrices of the form (E_{0}(v_{j}^{∗}v_{i})) whenever v_{1}, …, v_{n} ∈ G have height < N, then the n × n matrix whose ijth term is the right side of (7) must also be positive, because φ is a completely positive map and 0 ≤ φ(e) ≤ e. It follows from Lemma 3.3 that (E_{0}(u_{j}^{∗}u_{i})) must be a positive n × n matrix whenever u_{1}, …, u_{n} ∈ G have height ≤ N.

proof of Lemma 3.3. We identify the unit e of A with its image δ_{(0)}·e ∈ G. Fix i, 1 ≤ i ≤ n, and write u_{i}e = δ_{(n_{1},...,n_{k})} · a_{1} ⊗ ⋯ ⊗ a_{k}. Note that n_{k} must be 0 because u_{i} has been multiplied on the right by e.

If n_{1} > 0 we choose ℓ, 1 < ℓ < k such that n_{1}, n_{2}, …, n_{ℓ} are positive and n_{ℓ+1} = 0. Setting v_{i} = δ_{(n_{1}−1,...,n_{ℓ}−1)} · a_{1} ⊗ ⋯ ⊗ a_{ℓ} and w_{i} = δ_{(n_{ℓ+1},...,n_{k})} · a_{ℓ+1} ⊗ ⋯ ⊗ a_{k}, we obtain a factorization u_{i} = σ(v_{i})w_{i}, and we define b_{i} and c_{i} by b_{i} = E_{0}(w_{i}), c_{i} = 0. If n_{1} = 0 then u_{i}e cannot be factored in this way; still, we set v_{i} = e, and b_{i} = c_{i} = E_{0}(u_{i}). This defines b_{i}, c_{i} and v_{i}.

One now verifies (7) in cases: where both u_{i}e and u_{j}e factor into a product of the form σ(v)w, when one of them so factors and the other does not, and when neither does. For example, if u_{i}e = σ(v_{i})w_{i} and u_{j}e = σ(v_{j})w_{j} both factor, then we can make use of the formulas E_{0}(f) = eE_{0}(f)e = E_{0}(efe) for f ∈ ℓ^{1}(Σ), E_{0}(fg) = E_{0}(f)E_{0}(g) for f, g ∈ [Aℓ^{1}(Σ)A], and E_{0} ∘ σ = φ ∘ E_{0}, to write

E_{0}(u_{j}^{∗}u_{i}) = E_{0}(eu_{j}^{∗}u_{i}e) = E_{0}((u_{j}e)^{∗}u_{i}e) = E_{0}(w_{j}^{∗}σ(v_{j}^{∗}v_{i})w_{i})
= b_{j}^{∗}E_{0}(σ(v_{j}^{∗}v_{i}))b_{i} = b_{j}^{∗}φ(E_{0}(v_{j}^{∗}v_{i}))b_{i}.

If u_{i}e = σ(v_{i})w_{i} so factors and u_{j}e = eu_{j}e does not, then we write

E_{0}(u_{j}^{∗}u_{i}) = E_{0}((u_{j}e)^{∗}u_{i}e) = E_{0}(u_{j}e)^{∗}E_{0}(σ(v_{i})w_{i}) = b_{j}^{∗}E_{0}(σ(v_{i}))b_{i}
= b_{j}^{∗}φ(E_{0}(v_{i}))b_{i} = b_{j}^{∗}(φ(E_{0}(v_{j}^{∗}v_{i}))b_{i},

noting that in this case v_{j}^{∗} = e. A similar string of identities settles the case u_{i}e = eu_{i}e, u_{j}e = σ(v_{j})w_{j}.

Note that in each of the preceding three cases, the terms c_{j}^{∗}(e − φ(e))c_{i} were all 0.

In the remaining case where u_{i}e = eu_{i}e and u_{j}e = eu_{j}e, we can write E_{0}(u_{j}^{∗}u_{i}) = E_{0}(eu_{j}^{∗}u_{i}e) = E_{0}((eu_{j}e)^{∗}eu_{i}e) = E_{0}(eu_{j}e)^{∗}E_{0}(eu_{i}e) = b_{j}^{∗}b_{i}. Formula (7) persists for this case too, since v_{i} = v_{j} = e and we can write

b_{j}^{∗}b_{i} = b_{j}^{∗}φ(e)b_{i} + b_{j}^{∗}(e − φ(e))b_{i} = b_{j}^{∗}φ(E_{0}(v_{j}^{∗}v_{i}))b_{i} + c_{j}^{∗}(e − φ(e))c_{i}. □

4. the hierarchy of dilations

Let (A, φ) be a pair consisting of an arbitrary C^{∗}-algebra A and a completely positive linear map φ : A → A satisfying ∥φ∥ ≤ 1. Definition 4.1. A dilation of (A, φ) is an A-dynamical system (i, B, α) with the property that there is an α-expectation E : B → A satisfying

E(α(a)) = φ(a), a ∈ A.

Notice that the α-expectation E : B → A associated with a dilation of (A, φ) is uniquely determined, by Theorem 1.3. The class of all dilations of (A, φ) is contained in the class of all A-dynamical systems, and it is significant that it is also a subcategory. More explicitly, if (i_{1}, B_{1}, α_{1}) and (i_{1}, B_{2}, α_{2}) are two dilations of (A, φ), and if θ : B_{1} → B_{2} is a homomorphism of A-dynamical systems, then the respective α-expectations E_{1}, E_{2} must also transform consistently

E_{2} ∘ θ = E_{1}.  (8)

This follows from Theorem 2.3, since both E_{1} and E_{2} ∘θ are α_{1}-expectations that project α_{1}(a) to φ(a), a ∈ A.

Theorem 1.3 implies that every pair (A, φ) can be dilated to the universal A-dynamical system (i, PA, σ). Let E_{φ} : PA → A be the σ-expectation satisfying E_{φ}(σ(a)) = φ(a), a ∈ A. The preceding remarks imply that for every other dilation (i, B, α), there is a unique surjective ∗-homomorphism θ : PA → B such that θ ∘ σ = α ∘ θ, E ∘ θ = E_{φ}, and which fixes A elementwise. Thus, (i, PA, σ) is a universal dilation of (A, φ). The universal dilation is obviously too large, since its structure bears no relation to φ. Thus it is significant that there is a smallest (A, φ) dilation, whose structure is more closely tied to φ. We now discuss the basic properties of this minimal dilation; we examine its structure in section 5.

In general, every completely positive map of C^{∗}-algebras E : B_{1} → B_{2} gives rise to a norm-closed two-sided ideal ker E in B_{1} as follows

ker E = {x ∈ B_{1} : E(bxc) = 0, for all b, c ∈ B_{1}}.

In more concrete terms, if B_{2} ⊆ B(H) acts concretely on some Hilbert space and E(x) = V^{∗}π(x)V is a Stinespring decomposition of E, where π is a representation of B_{1} on some other Hilbert space K and V : H → K is a bounded operator such that π(B_{1})V H has K as its closed linear span, then one can verify that

ker E = {x ∈ B_{1} : π(x) = 0}.  (9)

Notice too that ker E = {0} iff E is faithful on ideals in the sense that for every two-sided ideal J ⊆ B, one has E(J) = {0} ⟹ J = {0}. Proposition 4.2. Let (i, B, α) be an A-dynamical system and let E : B → A be an α-expectation. Then ker E is an α-invariant ideal with the property A ∩ ker E = {0}.

If (i_{1}, B_{1}, α_{1}) and (i_{2}, B_{2}, α_{2}) are two dilations of (A, φ) and θ : B_{1} → B_{2} is a homomorphism of A-dynamical systems, then

ker E_{1} = {x ∈ B_{1} : θ(x) ∈ ker E_{2}}.  (10)

Proof. That A∩ker E = {0} is clear from the fact that if a ∈ A∩ker E then AaA = E(AaA) = {0}, hence a = 0. Relation (10) is also straightforward, since θ(B_{1}) = B_{2} and E_{2} ∘ θ = E_{1}. Indeed, for each x ∈ B_{1}, we have

E_{2}(B_{2}θ(x)B_{2}) = E_{2}(θ(B_{1})θ(x)θ(B_{1})) = E_{2}(θ(B_{1}xB_{1})) = E_{1}(B_{1}xB_{1}),

from which (10) follows.

To see that α(ker E) ⊆ ker E, choose k ∈ ker E. Since B is spanned by all finite products of elements α^{n}(a), a ∈ A, n = 0, 1, …, it suffices to show that E(yα(k)x) = 0 for all y ∈ B and all x of the form x = α^{m_{1}}(a_{1}) ⋯ α^{m_{k}}(a_{k}). Being a completely positive contraction, E satisfies the Schwarz inequality

E(yα(k)x)^{∗}E(yα(k)x) ≤ E(x^{∗}α(k^{∗})y^{∗}yα(k)x) ≤ ∥y∥^{2}E(x^{∗}α(k^{∗}k)x);

hence it suffices to show that E(x^{∗}α(k^{∗}k)x) = 0. To prove the latter, one can argue cases as follows. Assuming that m_{i} > 0 for all i, then x = α(x_{0}) for some x_{0} ∈ B, and using E ∘ α = φ ∘ E one has

E(x^{∗}α(k^{∗}k)x) = E(α(x_{0}k^{∗}kx_{0})) = φ(E(x_{0}^{∗}k^{∗}kx_{0})) = 0.

For the remaining case where some m_{i} = 0, notice that x must have one of the forms x = a ∈ A (when all m_{i} are 0), or x = ax_{0} with x_{0} ∈ B (when m_{1} = 0 and some other m_{j} is positive), or α(x_{1})ax_{2} with a ∈ A, x_{1}, x_{2} ∈ B (when m_{1} > 0, …, m_{r−1} > 0 and m_{r} = 0), and in each case E(x^{∗}α(k^{∗}k)x) = 0. If x = α(x_{1})ax_{2}, for example, then we make use of Formula (6) to write

E(x^{∗}α(k^{∗}k)x) = E(x_{2}^{∗}a^{∗}α(x_{1}^{∗}k^{∗}kx_{1})ax_{2}) = E(x_{2})^{∗}a^{∗}E(α(x_{1}^{∗}k^{∗}kx_{1})aE(x_{2}).

The term on the right vanishes since E(α(x_{1}^{∗}k^{∗}kx_{1}) = φ(E(x_{1}^{∗}k^{∗}kx_{1})) = 0. The other cases are dealt with similarly, and α(ker E) ⊆ ker E follows. □

We deduce the existence of minimal dilations and their basic characterization as follows. Fix a pair (A, φ), where φ : A → A is a completely positive contraction, let σ be the shift on PA, and let E_{φ} : PA → A be the unique σ-expectation satisfying E_{φ}(σ(a)) = φ(a), a ∈ A. Proposition 4.2 implies that σ leaves ker E_{φ} invariant, thus it can be promoted to an endomorphism σ̇ of the quotient C^{∗}-algebra PA/ ker E_{φ}. Moreover, since A ∩ ker E_{φ} = {0}, the inclusion of A in PA promotes to an inclusion of A in PA/ ker E_{φ}. Thus we obtain an A-dynamical system (i, PA/ ker E_{φ}, σ̇) having a natural σ̇-expectation Ė defined by Ė(x + ker E_{φ}) = E_{φ}(x) + ker E_{φ}, which satisfies Ė(σ̇(a)) = φ(a), a ∈ A. It is called the minimal dilation of (A, φ) in light of the following: Corollary 4.3. The dilation (i, PA/ ker E_{φ}, σ̇) of (A, φ) has the following properties.

(1) (i, PA/ ker E_{φ}, σ̇) is subordinate to all other dilations of (A, φ).

(2) The σ̇-expectation Ė of (i, PA/ ker E_{φ}, σ̇) satisfies ker Ė = {0}.

(3) Every dilation (i, B, α) of (A, φ) whose α-expectation E satisfies

ker E = {0} is isomorphic to (i, PA/ ker E_{φ}, σ̇). Proof. (2) follows by construction of (i, PA/ ker E_{φ}, σ̇), since the kernel ideal of its σ̇-expectation has been reduced to {0}.

To prove (1), let (i, B, α) be an arbitrary dilation of (A, φ). By the universal property of (i, PA, σ) there is a surjective ∗-homomorphism θ : PA → B satisfying θ ∘ σ = α ∘ θ; and by (8) one has E ∘ θ = E_{φ}. Formula (10) implies that ker E_{φ} contains ker θ, hence we can define a morphism of C^{∗}-algebras ω : B → PA/ ker E_{φ} by way of ω(θ(x)) = x + ker E_{φ}, for all x ∈ PA. Obviously, ω is a homomorphism of A-dynamical systems, and we conclude that (i, B, α) ≥ (i, PA/ ker E_{φ}, σ̇).

For (3), notice that if (i, B, α) is a dilation of (A, φ) whose α-expectation E : B → A satisfies ker E = {0} and θ : PA → B is the homomorphism of the previous paragraph, then Formula (10) implies that ker E_{φ} = ker θ. Thus ω : B → PA/ ker E_{φ} has trivial kernel, hence it must implement an isomorphism of A-dynamical systems (i, B, α) ≅ (i, PA/ ker E_{φ}, σ̇). □

5. structure of minimal dilations

Corollary 4.3 implies that minimal dilations of (A, φ) exist for every contractive completely positive map φ : A → A, and that they are characterized by the fact that their α-expectations are faithful on ideals. The latter imposes strong requirements on the structure of minimal dilations, and we conclude by elaborating on these structural issues. Definition 5.1. A standard dilation of (A, φ) is a dilation (i, B, α) such that A = pBp is an essential corner of B whose projection p ∈ M(B) satisfies pα(x)p = φ(pxp), x ∈ B.

In such cases, E(x) = pxp is the α-expectation of B on A. Standard dilations are most transparent in the special case where A has a unit e and φ(e) = e. To illustrate that, let B be a C^{∗}-algebra containing A and let α be an endomorphism of B with the following property:

φ(a) = eα(a)e,  a ∈ A,  (11)

e denoting the unit of A. We may also assume that B is generated by A ∪ α(A) ∪ α^{2}(A) ∪ ⋯, so that (i, B, α) becomes an A-dynamical system.

Proposition 5.2. The projection e ∈ B satisfies α(e) ≥ e, A = eBe is a hereditary subalgebra of B, and the map E(x) = exe defines an α-expectation from B to A. If, in addition, A is an essential subalgebra of B, then (i, B, α) is a standard dilation of (A, φ).

Sketch of proof. Formula (11) implies that eα(e)e = φ(e) = e, hence α(e) ≥ e. It follows immediately that eα(exe)e = eα(x)e for x ∈ B.

At this point, a simple induction establishes eα^{n}(a)e = φ^{n}(a), a ∈ A, n = 0, 1, 2, . . .. An argument similar to the proof of Theorem 2.3 allows one to evaluate more general expectation values as in (5)

eα^{n_{1}}(a_{1})α^{n_{2}}(a_{2}) ⋯ α^{n_{k}}(a_{k})e = [n_{1}, …, n_{k}; a_{1}, …, a_{k}],

which implies eBe ⊆ A. Hence A = eBe is a hereditary subalgebra of B.

With these formulas in hand one finds that the conditional expectation E(x) = exe satisfies axioms E1 and E2 of Definition 4.1. Hence (i, B, α) is a standard dilation of (A, φ) whenever A is an essential subalgebra of B. □

We remark that the converse is also true: given (A, φ) for which A has a unit e and φ(e) = e, then every standard dilation has the properties of Proposition (5.2). The description of standard dilations in general, where A is unital and ∥φ∥ ≤ 1 or is perhaps nonunital, becomes more subtle.

The universal dilation (i, PA, σ) of (A, φ) is not a standard dilation. For example, when A has a unit e one can make use of the universal property of PA = A ∗ A ∗ ⋯ to exhibit representations π : PA → B(H) such that π(e) and π(σ(e)) are nontrivial orthogonal projections. Hence σ(e) ≱ e. Moreover, A is not a hereditary subalgebra of PA, and the conditional expectation of Theorem 1.3 is never of the form x ↦ exe.

On the other hand, we now show that minimal dilations of (A, φ) must be standard. This is based on the following characterization of essential corners in terms of conditional expectations.

Proposition 5.3. For every inclusion of C^{∗}-algebras A ⊆ B, the following are equivalent.

(i) A is an essential corner pBp of B.

(ii) There is a conditional expectation E : B → A whose restriction to [ABA] is multiplicative, and which satisfies ker E = {0}.

Moreover, the conditional expectation E : B → A of (ii) is the compression map E(x) = pxp, and it is unique. The projection p ∈ M(B) satisfies

lim_{n→∞} ∥xe_{n} − xp∥ = 0, x ∈ B,

where e_{n} is any approximate unit for A, and it defines the closed left ideal generated by A as follows: [BA] = Bp. Proof. The implication (i) ⟹ (ii) is straightforward, since the compression map E(x) = pxp obviously defines a conditional expectation of B on A = pBp that is multiplicative on [ABA]. If x ∈ B satisfies E(BxB) = {0} then pBx^{∗}xBp = {0}, hence xBA = xBpBp = {0}, and therefore x = 0 because [BAB] is assumed to be an essential ideal in (i).

(ii) ⟹ (i). Given a conditional expectation E : B → A satisfying (ii), we may assume that A ⊆ B(H) acts nondegenerately on some Hilbert space (e.g., represent B faithfully on some Hilbert space and take H to be the closed linear span of the ranges of all operators in A). Thus E : B → B(H) becomes an operator-valued completely positive map of norm 1, having a Stinespring decomposition E(x) = V^{∗}π(x)V, with π a representation of B on a Hilbert space K, and V : H → K a contraction with [π(B)V H] = K.

Let P be the projection on [π(A)K]. We claim that V is an isometry with V V^{∗} = P. To prove that, choose a ∈ A, b ∈ B, and let e_{n} be an approximate unit for A. Since E is multiplicative on [ABA] we can write

V^{∗}π(b^{∗}a^{∗})(V V^{∗} − 1)π(ab)V^{∗} = E(b^{∗}a)E(ab) − E(b^{∗}a^{∗}ab) =
lim_{n} e_{n}(E(b^{∗}a^{∗})E(ab) − E(b^{∗}a^{∗}ab))e_{n} =
lim_{n}(E(e_{n}b^{∗}a^{∗})E(abe_{n}) − E(e_{n}b^{∗}a^{∗}abe_{n})) = 0.

It follows that V V^{∗} − 1 vanishes on the closed linear span of π(A)π(B)V H, namely [π(A)K]; hence V V^{∗} ≥ P. On the other hand, for a ∈ A we have V a = V E(a) = V V^{∗}π(a)V = π(a)V. Thus V H ⊆ [V AH] = [π(A)V H] ⊆ P K; hence V V^{∗} ≤ P. That V is an isometry follows from the fact that for a ∈ A, V^{∗}V a = V^{∗}V E(a) = V^{∗}V V^{∗}π(a)V = V^{∗}π(a)V = E(a) = a, and by nondegeneracy H is the closed linear span of {aξ : a ∈ A, ξ ∈ H}.

We claim that P = V V^{∗} belongs to the multiplier algebra of π(B). For that, choose an approximate unit e_{n} for A. Since both π(e_{n}) and V V^{∗} are self-adjoint, it suffices to show that for every b ∈ B, π(b)π(e_{n}) → π(b)V V^{∗} in norm as n → ∞. Using V V^{∗}π(e_{n}) = π(e_{n})V V^{∗} = π(e_{n}), we can write

∥π(b)(π(e_{n}) − V V^{∗})∥^{2} = ∥(π(e_{n}) − V V^{∗})π(b^{∗}b)(π(e_{n}) − V V^{∗})∥ =
∥V V^{∗}(π(e_{n}b^{∗}be_{n}) − π(e_{n}b^{∗}b) − π(b^{∗}be_{n}) + π(b^{∗}b))V V^{∗}∥ =
∥V (E(e_{n}b^{∗}be_{n}) − E(e_{n}b^{∗}b) − E(b^{∗}be_{n}) + E(b^{∗}b))V^{∗}∥ ≤
∥e_{n}E(b^{∗}b)e_{n} − e_{n}E(b^{∗}b) − E(b^{∗}b)e_{n} + E(b^{∗}b)∥,

and the last term tends to 0 as n → ∞ because e_{n} is an approximate unit for A and E(b^{∗}b) ∈ A. It follows that π(B)V V^{∗} = π(A)P is the closed left ideal in π(B) generated by π(A).

We claim next that π(A) = P π(B)P is a corner of π(B). Indeed,

V V^{∗}π(B)V V^{∗} = V E(B)V^{∗} = π(E(B))V V^{∗} = π(A)V V^{∗} = π(A).

It is essential because for any operator T ∈ B(K) for which T π(BA) = {0} we must have T π(BA)V V^{∗} = {0}. But since K is spanned by vectors of the form π(b)V aξ = π(b)π(a)V ξ for a ∈ A, b ∈ B, the only possibility is T = 0.

Finally, π must be a faithful representation because π(x) = 0 implies

E(BxB) = V^{∗}π(B)π(x)π(B)V = {0},

and the latter implies x = 0 by hypothesis (ii). The preceding assertions can now be pulled back through the isomorphism π : B → π(B) to give (i). □

Combining Corollary 4.3 with Proposition 5.3, we obtain: Theorem 5.4. For every (A, φ) as above, the minimal dilation of (A, φ) is a standard dilation satisfying the assertions of Proposition 5.3. All standard dilations of (A, φ) are equivalent to the minimal one.

We remark that Theorem 5.4, together with a theorem of Larry Brown [Bro77], implies that the C^{∗}-algebra B of the minimal dilation (i, B, α) of (A, φ) can be embedded in the multiplier algebra of A ⊗ K.

Concluding Remarks. It is appropriate to review some highlights of the literature on noncommutative dilation theory, since it bears some relationship to the contents of §§4–5. Several approaches to dilation theory for semigroups of completely positive maps have been proposed since the mid 1970s, including work of Evans and Lewis [EL77], Accardi et al [AL82], Kümmerer [Küm85], Sauvageot [Sau86], and many others. Our attention was drawn to these developments by work of Bhat and Parthasarathy [BP94], in which the first dilation theory for CP semigroups acting on B(H) emerged that was effective for our work on E_{0}-semigroups [Arv97], [Arv00]. SeLegue [SeL97] showed how to apply multi-operator dilation theory to obtain the Bhat–Parthasarathy results, and he calculated the expectation values of the n-point functions of such dilations. Recently, Bhat and Skeide [BS00] have initiated an approach to the subject that is based on Hilbert modules over C^{∗}-algebras and von Neumann algebras.

We intend to take up applications to semigroups of completely positive maps elsewhere.

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[SeL97] Dylan SeLegue. Minimal Dilations of CP Maps and a C^{∗}-Extension of the Szegö Limit. PhD thesis, University of California, Berkeley, June 1997.

Department of Mathematics, University of California, Berkeley, CA 94720

E-mail address: arveson@mail.math.berkeley.edu