Generators of Noncommutative Dynamics

15 Pages·0.21 MB·English

by  William Arveson

GENERATORS OF NONCOMMUTATIVE DYNAMICS
2
0 WILLIAM ARVESON
0
2
Abstract. For a fixed C∗-algebra A, we consider all noncommutative
n
dynamical systems that can be generated by A. More precisely, an A-
a
J dynamicalsystem isatriple(i,B,α)whereαisa∗-endomorphism ofa
C∗-algebra B, and i : A ⊆ B is the inclusion of A as a C∗-subalgebra
5
withthepropertythatB isgeneratedbyA∪α(A)∪α2(A)∪···. There
1
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
A establish certain properties of (i,PA,α) and give applications to some
O concrete issues of noncommutativedynamics.
Forexample,weshowthateverycontractivecompletelypositivelin-
.
h earmapϕ:A→AgivesrisetotoauniqueA-dynamicalsystem(i,B,α)
t thatis“minimal”withrespecttoϕ,andweshowthatitsC∗-algebraB
a
m can be embedded in themultiplier algebra of A⊗K.
[

1
v 1. Generators
7
3 The flow of time in quantum theory is represented by a one-parameter
1 ∗
group of ∗-automorphisms {α : t ∈ R} of a C -algebra B. There is often
1 t
∗
0 a C -subalgebra A ⊆ B that can be singled out from physical considera-
2 tions which, together with its time translates, generates B. For example,
0
in nonrelativistic quantum mechanics the flow of time is represented by a
/
h one-parameter group of automorphisms of B(H), and the set of all bounded
t
a continuous functions of the configuration observables at time 0 is a com-
m mutative C∗-algebra A. The set of all time translates α (A) of A generates
t
∗
: an irreducible C -subalgebra B of B(H). In particular, for different times
v
∗
t 6= t , theC -algebras α (A)andα (A)donotcommutewitheach other.
i 1 2 t1 t2
X
Indeed, no nontrivial relations appear to exist between α (A) and α (A)
t1 t2
r when t 6= t .
a 1 2
In this paper we look closely at this phenomenon, in a simpler but anal-
∗
ogous 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
∗
∗-endomorphismαactingonaC -algebraB andaninjective∗-homomorphism
i: A→ B, such that B is generated by i(A)∪α(i(A))∪α2(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
1991 Mathematics Subject Classification. 46L55, 46L09.
Partially supported by NSFgrants DMS-9802474 and DMS-0100487.
1
2 WILLIAM ARVESON
a specified way, with the property that B is the norm-closed linear span of
finite products of the following form
B = span{αn1(a )αn2(a )···αnk(a )} (1)
1 2 k
where n ,...,n ≥ 0, a ,...,a ∈ A, k = 1,2,...
1 k 1 k
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 mem-
bers. 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, de-
finedby(i ,B ,α )≥ (i ,B ,α )iffthereisa∗-homomorphismθ :B → B
1 1 1 2 2 2 1 2
satisfying θ◦α = α ◦θ and θ(a)= a for a∈ A. Since θ fixes A, it follows
1 2
from (1) that θ must be surjective, θ(B ) = B , hence (i ,B ,α ) is a quo-
1 2 2 2 2
tient of (i ,B ,α ). Two A-dynamical systems are said to be equivalent if
1 1 1
∗
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 representa-
tives 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 ,θ ,... of A into the C -
0 1
algebra PA such PA is generated by θ (A) ∪ θ (A) ∪ ··· and such that
0 1
the following universal property is satisfied: for every sequence π ,π ,...
0 1
∗
of ∗-homomorphisms of A into some other C -algebra B, there is a unique
∗-homomorphism ρ : PA → B such that π = ρ◦θ , k = 0,1,.... Nonde-
k k
generate representations of PA correspond to sequences π¯ = (π ,π ,...) of
0 1
representations π : A→ B(H) of A on a common Hilbert space H, subject
k
to no condition other than the triviality of their common nullspace
ξ ∈ H, π (A)ξ = {0}, k = 0,1,··· =⇒ ξ = 0.
k
A simple argument establishes the existence of PA by taking the direct sum
of a sufficiently large set of such representation sequences π¯.
This definition does not exhibit PA in concrete terms (see §3 for that),
butitdoesallow ustodefineauniversalA-dynamicalsystem. Theuniversal
property of PA implies that there is a shift endomorphism σ : PA → PA
defined uniquely by σ ◦θ = θ , k = 0,1,.... It is quite easy to verify
k k+1
that θ is an injective ∗-homomorphism of A in PA, and we use this map
0
to identify A with θ (A) ⊆ PA. Thus the triple (i,PA,σ) becomes an A-
0
dynamical 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
GENERATORS OF NONCOMMUTATIVE DYNAMICS 3
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 kEk = 1.
Definition 1.2. Let(i,B,α) bean A-dynamical system. Anα-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 kϕk ≤ 1. Axiom E1 makes the assertion
E◦α =ϕ◦E. (2)
where ϕ = E ◦α ↾ is the linear map of A associated with E.
A
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 sec-
tion, see Theorem 2.3. Existence is taken up in §3, see Theorem 3.2.
4 WILLIAM ARVESON
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 n¯ = (n ,...,n )
1 k
a ,...,a ∈A 7→ [n¯;a ,...,a ] ∈ A
1 k 1 k
is a k-linear mapping, all of which satisfy
MP1. ϕ([n¯;a ,...,a ]) = [n +1,n +1,...,n +1;a ,...,a ].
1 k 1 2 k 1 k
MP2. Given a k-tuple for which n = 0 for some ℓ between 1 and k,
ℓ
[n¯;a1,...,ak] = [n1,...,nℓ−1;a1,...,aℓ−1]aℓ[nℓ+1,...,nk;aℓ+1,...,ak].
Remark 2.2. The proofs of both existence and uniqueness are straightfor-
ward arguments using induction on the number k of variables, and we omit
them. Notethatinthesecondaxiom MP2, wemakethenaturalconventions
when ℓ has one of the extreme values 1, k. For example, if ℓ = 1, then MP2
should be interpreted as
[0,n ,...,n ;a ,...,a ]= a [n ,...,n ;a ,...,a ].
2 k 1 k 1 2 k 2 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]= ϕn(a), a∈ A, n = 0,1,....
One may calculate any particular moment polynomial explicitly, but the
computationsquicklybecomeatediousexerciseinthearrangementofparen-
theses. For example,
[2,6,3,4;a,b,c,d] = ϕ2(aϕ(ϕ3(b)cϕ(d))),
[6,4,2,3;a,b,c,d] = ϕ2(ϕ2(ϕ2(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 ,...,n ;a ,...,a ] = [n ,...,n ;a ,...,a ]. (4)
1 k 1 k k 1 k 1
Indeed, one finds that the sequence of polynomials [[·;·]] defined by
∗ ∗ ∗
[[n ,...,n ;a ,...,a ]] = [n ,...,n ;a ,...,a ]
1 k 1 k k 1 k 1
also satisfies axioms MP1 and MP2, and hence must coincide with the mo-
ment 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.
GENERATORS OF NONCOMMUTATIVE DYNAMICS 5
Theorem 2.3. Let ϕ :A → A be a completely positive map on A, satisfying
kϕk ≤ 1, with associated moment polynomials [n ,...,n ;a ,...,a ].
1 k 1 k
Let (i,B,α) be an A-dynamical system and let E : B → A be an α-
expectation with the property E(α(a)) = ϕ(a), a ∈A. Then
E(αn1(a )αn2(a )···αnk(a )) = [n ,...,n ;a ,...,a ]. (5)
1 2 k 1 k 1 k
for every k = 1,2,..., n ≥ 0, a ∈ A. In particular, there is at most one
k k
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 polyno-
mials [[·;·]] defined by
[[n ,...,n ;a ,...,a ]] = E(αn1(a )···αnk(a ))
1 k 1 k 1 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 for A we can write E(xay) as follows:
n
lim e E(xae y)e = lim E(e xae ye ) = lim E(e xa)E(e ye )
n n n n n n n n n
n→∞ n→∞ n→∞
= lim e E(x)ae E(y)e = E(x)aE(y).
n n n
n→∞
Thus formula (5) follows from the uniqueness assertion of Proposition 2.1.
Theuniquenessoftheα-expectation associated withϕisnowapparentfrom
formulas (5) and (1).
3. Existence of α-expectations
In this section we show that every completely positive map ϕ : A → A,
with kϕk ≤ 1, gives rise to an α-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 ∗-
algebraℓ1(Σ),insuchawaythatthedesiredconditionalexpectationappears
as a completely positive map on ℓ1(Σ). The details are as follows.
Let S be the set of finite sequences n¯ = (n ,n ,...,n ) of nonnegative
1 2 k
integers, k = 1,2,... which have distinct neighbors,
n1 6= n2,n2 6= n3,...,nk−1 6=nk.
MultiplicationandinvolutionaredefinedinS asfollows. Theproductoftwo
elements m¯ = (m ,...,m ),n¯ = (n ,...,n ) ∈ S is defined by conditional
1 k 1 ℓ
concatenation
(m ,...,m ,n ,...,n ), if m 6= n ,
1 k 1 ℓ k 1
m¯ ·n¯ =
((m1,...,mk,n2,...,nℓ), if mk = n1,
where we make the natural conventions when n¯ = (q) is of length 1, namely
m¯ · (q) = (m ,...,m ,q) if m 6= q, and m¯ · (q) = m¯ if m = q. The
1 k k k
involution in S is defined by reversing the order of components
∗
(m ,...,m ) = (m ,...,m ).
1 k k 1
6 WILLIAM ARVESON
One finds that S is an associative ∗-semigroup.
∗
Fixing a C -algebra A, we attach a Banach space Σ to every k-tuple
ν
ν = (n ,...,n ) ∈ S as follows
1 k
Σ = A⊗ˆ ···⊗ˆA,
ν
k times
the k-fold projective tensor product of copies of the Banach space A. We
| {z }
assemble the Σ into a family of Banach spaces over S, p : Σ → S, by way
ν
of Σ = {(ν,ξ) : ν ∈ S,ξ ∈ E }, p(ν,ξ) = ν.
ν
We introduce a multiplication in Σ as follows. Fix µ = (m ,...,m ) and
1 k
ν = (n ,...,n ) in S and choose ξ ∈ Σ , η ∈ Σ . If m 6= n then ξ ·η is
1 ℓ µ ν k 1
defined as the tensor product ξ⊗η ∈Σµ·ν. If mk = n1 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 Σµ⊗AΣν onto Σµ·ν by making
identifications of elementary tensors as follows:
(a1⊗···⊗ak)⊗A(b1⊗···⊗bℓ) ∼ a1⊗···⊗ak−1⊗akb1⊗b2⊗···⊗bℓ.
With this convention ξ·η is defined by
ξ·η = ξ⊗Aη ∈ Σµ·ν.
This defines an associative multiplication in the family of Banach spaces Σ.
ThereisalsoanaturalinvolutioninΣ,definedoneachΣ ,µ = (m ,...,m )
µ 1 k
as the unique antilinear isometry to Σµ∗ satisfying
∗ ∗ ∗
((m ,...,m ),a ⊗···⊗a ) = ((m ,...,m ),a ⊗···⊗a ).
1 k 1 k k 1 k 1
This defines an isometric antilinear mapping of the Banach space Σ onto
µ
Σµ∗, for each µ ∈ S, and thus the structure Σ becomes an involutive ∗-
semigroup in which each fiber Σ is a Banach space.
µ
Let ℓ1(Σ) be the Banach ∗-algebra of summable sections. The norm and
∗ ∗ ∗
involutionarethenaturaloneskfk = kf(µ)k,f (µ) = f(µ ) . Noting
µ∈Σ
that Σλ·Σµ ⊆ Σλ·µ, the multiplicatioPn in ℓ1(Σ) is defined by convolution
f ∗g(ν) = f(λ)·g(µ),
λ·µ=ν
X
and one easily verifies that ℓ1(Σ) is a Banach ∗-algebra.
For µ = (m ,...,m ) ∈ S and a ,...,a ∈ A we define the function
1 k 1 k
δ ·a ⊗···⊗a ∈ ℓ1(Σ)
µ 1 k
to be zero except at µ, and at µ it has the value a ⊗···⊗a ∈ Σ . These
1 k µ
elementary functions have ℓ1(Σ) as their closed linear span. Finally, there
is a natural sequence of ∗-homomorphisms θ ,θ ,··· : A→ ℓ1(Σ) defined by
0 1
θ (a) = δ ·a, a ∈ A, k = 0,1,...,
k (k)
and these maps are related to the generating sections by
δ ·a ⊗···⊗a = θ (a )θ (a )···θ (a ).
(n1,...,nk) 1 k n1 1 n2 2 nk k
GENERATORS OF NONCOMMUTATIVE DYNAMICS 7
The algebra ℓ1(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 : A → B(H), k = 0,1,..., fix ν = (n ,...,n ) ∈ S. There is a unique
k 1 k
bounded linear operator L : Σ → B(H) of norm 1 that is defined by its
ν ν
action on elementary tensors as follows
L (a ⊗···⊗a ) = π (a )···π (a ).
ν 1 k n1 1 nk k
Thus there is a bounded linear map π˜ : ℓ1(Σ)→ B(H) defined by
π˜(f)= L (f(µ)), f ∈ ℓ1(Σ).
µ
µ∈S
X
One finds that π˜ is a ∗-representation of ℓ1(Σ) with kπ˜k = 1. This repre-
sentation satisfies π˜ ◦θ = π , k = 0,1,2,,.... Conversely, every bounded
k k
∗-representation π˜ of ℓ1(Σ) on a Hilbert space H is associated with a se-
quence of representations π ,π ,... of A on H by way of π = π˜◦θ .
0 1 k k
The results of the preceding discussion are summarized as follows:
Proposition 3.1. The enveloping C∗-algebra C∗(ℓ1(Σ)), together with the
sequence of homomorphisms θ˜ ,θ˜ ,··· : A→ C∗(ℓ1(Σ)) defined by the maps
0 1
θ ,θ ,··· : A → ℓ1(Σ), has the same universal property as the infinite free
0 1
product PA =A∗A∗···, and is therefore isomorphic to PA.
Notice that the natural shift endomorphism of ℓ1(Σ) is defined by
σ :δ ·ξ 7→ δ ·ξ, ν = (n ,...,n ) ∈ Σ, ξ ∈ Σ
(n1,...,nk) (n1+1,...,nk+1) 1 k ν
anditpromotestothenaturalshiftendomorphismofPA = C∗(ℓ1(Σ)). The
inclusion of A in ℓ1(Σ) is given by the map θ (a) = δ a∈ ℓ1(Σ), and it too
0 (0)
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 : ℓ1(Σ) → A satisfying
0
E (δ ·a ⊗···⊗a )= [n ,...,n ;a ,...,a ],
0 (n1,...,nk) 1 k 1 k 1 k
for (n ,...,n ) ∈ S, a ,...,a ∈ A, k = 1,2,..., and kE k = kϕk ≤ 1.
1 k 1 k 0
Using the axioms MP1 and MP2, one finds that the map E preserves the
0
adjoint (see Equation (4)), satisfies the conditional expectation property
E (af) = aE (f) for a ∈ A, f ∈ ℓ1(Σ), that the restriction of E to the
0 0 0
“hereditary” ∗-subalgebra of ℓ1(Σ) spanned by θ (A)ℓ1(Σ)θ (A) is multi-
0 0
plicative, and that it is related to ϕ by E ◦σ = ϕ◦E andE (σ(a)) = ϕ(a),
0 0 0
a ∈ A. Thus, E satisfies the axioms of Definition 1.2, suitably interpreted
0
for the Banach ∗-algebra ℓ1(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 reduces to:
0
8 WILLIAM ARVESON
Theorem 3.2. For every n ≥ 1, a ,...,a ∈ A, and f ,...,f ∈ ℓ1(Σ), we
1 n 1 n
have
n
∗ ∗
a E (f f )a ≥ 0.
j 0 j i i
i,j=1
X
Consequently, E extends uniquely through the completion map ℓ1(Σ)→ PA
0
to a completely positive map E : PA → A that becomes a σ-expectation
ϕ
satisfying Equation (3).
We sketch the proof of Theorem 3.2, detailing the critical steps. Using
the fact that ℓ1(Σ) is spanned by the generating family
G = {δ ·a ⊗···⊗a :(n ,...,n )∈ S, a ,...,a ∈ A, k ≥ 1}
(n1,...,nk) 1 k 1 k 1 k
one easily reduces the proof of Theorem 3.2 to the following more concrete
assertion: for any finite set of elements u ,...,u in G, the n×n matrix
1 n
∗
(a )= (E (u u )) ∈ M (A) is positive.
ij 0 j i n
The latter is established by an inductive argument on the “maximum
height”max(h(u ),...,h(u )),wheretheheightofanelementu= δ ·
1 n (n1,...,nk)
a ⊗···⊗a inGisdefinedash(u) = max(n ,...,n ). Thegeneralcaseeas-
1 k 1 k
ily 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 ,...,u ∈ G such that the maximum height N =
1 n
max(h(u ),...,h(u )) is positive. For k = 1,...,n there are elements
1 n
b ,c ∈ A and v ∈ G such that h(v ) < N and
k k k k
∗ ∗ ∗ ∗
E (u u ) = b ϕ(E (v v ))b +c (e−ϕ(e))c , 1 ≤ i,j ≤n. (7)
0 j i j 0 j i i j i
Remark 3.4. Note that if an inductive hypothesis provides positive n×n
∗
matrices of the form (E (v v )) whenever v ,...,v ∈ G have height < N,
0 j i 1 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 (u u )) must be a positive n× n matrix
0 j i
whenever u ,...,u ∈G have height ≤ N.
1 n
proof of Lemma 3.3. WeidentifytheuniteofAwithitsimageδ ·e ∈ G.
(0)
Fix i, 1 ≤ i ≤ n, and write u e= δ ·a ⊗···⊗a . Note that n must
i (n1,...,nk) 1 k k
be 0 because u has been multiplied on the right by e.
i
If n > 0 we choose ℓ, 1 < ℓ < k such that n ,n ,...,n are positive and
1 1 2 ℓ
n = 0. Setting v = δ ·a ⊗···⊗a and w = δ ·
ℓ+1 i (n1−1,...,nℓ−1) 1 ℓ i (nℓ+1,...,nk)
a ⊗···⊗a , we obtain a factorization u = α(v )w , and we define b and
ℓ+1 k i i i i
c by b = E (w ), c = 0. If n = 0 then u e cannot be factored in this way;
i i 0 i i 1 i
still, we set v = e, and b = c = E (u ). This defines b , c and v .
i i i 0 i i i i
Onenowverifies(7)incases: wherebothu eandu efactorintoaproduct
i j
of the form α(v)w, when one of them so factors and the other does not, and
when neither does. For example, if u e = α(v )w and u e = α(v )w both
i i i j j j
factor,thenwecanmakeuseoftheformulasE (f)= eE (f)e = E (efe)for
0 0 0
GENERATORS OF NONCOMMUTATIVE DYNAMICS 9
f ∈ ℓ1(Σ), E (fg) = E (f)E (g) for f,g ∈ [Aℓ1(Σ)A], and E ◦α = ϕ◦E ,
0 0 0 0 0
to write
∗ ∗ ∗ ∗ ∗
E (u u ) = E (eu u e) = E ((u e) u e) = E (w α(v v )w )
0 j i 0 j i 0 j i 0 j j i i
∗ ∗ ∗ ∗
= b E (α(v v ))b = b ϕ(E (v v ))b .
j 0 j i i j 0 j i i
If u e = α(v )w so factors and u e = eu e does not, then we write
i i i j j
∗ ∗ ∗ ∗
E (u u ) = E ((u e) u e) = E (u e) E (α(v )w )= b E (α(v ))b
0 j i 0 j i 0 j 0 i i j 0 i i
∗ ∗ ∗
= b ϕ(E (v ))b = b (ϕ(E (v v ))b ,
j 0 i i j 0 j i i
∗
noting that in this case v =e. A similar string of identities settles the case
j
u e= eu e, u e = α(v )w .
i i j j j
∗
Note that in each of the preceding three cases, the terms c (e−ϕ(e))c
j i
were all 0.
In the remaining case where u e = eu e and u e = eu e, we can write
i i j j
∗ ∗ ∗ ∗ ∗
E (u u ) = E (eu u e) = E ((eu e) eu e) = E (eu e) E (eu e) = b b .
0 j i 0 j i 0 j i 0 j 0 i j i
Formula (7) persists for this case too, since v = v = e and we can write
i j
∗ ∗ ∗ ∗ ∗ ∗
b b = b ϕ(e)b +b (e−ϕ(e))b = b ϕ(E (v v ))b +c (e−ϕ(e))c .
j i j i j i j 0 j i i j i
4. the hierarchy of dilations
∗
Let (A,ϕ) be a pair consisting of an arbitrary C -algebra A and a com-
pletely positive linear map ϕ :A → A satisfying kϕk ≤ 1.
Definition 4.1. Adilationof(A,ϕ)isanA-dynamicalsystem(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 ,B ,α ) and
1 1 1
(i ,B ,α )aretwodilationsof(A,ϕ),andifθ :B → B isahomomorphism
1 2 2 1 2
ofA-dynamicalsystems,thentherespectiveα-expectationsE ,E mustalso
1 2
transform consistently
E ◦θ = E . (8)
2 1
Thisfollows fromTheorem2.3, sincebothE and E ◦θ areα -expectations
1 2 1
that project α (a) to ϕ(a), a∈ A.
1
Theorem 1.3 implies that every pair (A,ϕ) can bedilated to the universal
A-dynamical system (i,PA,σ). Let E : PA → A be the σ-expectation
ϕ
satisfying E (σ(a)) = ϕ(a), a ∈ A. The preceding remarks imply that for
ϕ
every other dilation (i,B,α), there is a unique surjective ∗-homomorphism
θ : PA → B such that θ ◦ σ = α ◦ θ, E ◦ θ = E , and which fixes A
ϕ
elementwise. Thus,(i,PA,σ) is a universal dilation of(A,ϕ). Theuniversal
dilationisobviouslytoolarge,sinceitsstructurebearsnorelationtoϕ. Thus
it is significant that there is a smallest (A,ϕ) dilation, whose structure is
10 WILLIAM ARVESON
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 → B
1 2
gives rise to a norm-closed two-sided ideal kerE in B as follows
1
kerE ={x ∈ B :E(bxc) = 0, for all b,c ∈ B }.
1 1
In more concrete terms, if B ⊆B(H) acts concretely on some Hilbert space
2
∗
and E(x) = V π(x)V is a Stinespring decomposition of E, where π is a
representation of B on some other Hilbert space K and V : H → K is a
1
bounded operator such that π(B )VH has K as its closed linear span, then
1
one can verify that
kerE = {x∈ B :π(x) = 0}. (9)
1
Notice too that kerE = {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 kerE is an α-invariant ideal with the property
A∩kerE = {0}.
If (i ,B ,α ) and (i ,B ,α ) are two dilations of (A,ϕ) and θ :B → B
1 1 1 2 2 2 1 2
is a homomorphism of A-dynamical systems, then
kerE = {x ∈ B : θ(x)∈ kerE }. (10)
1 1 2
Proof. ThatA∩kerE = {0}isclearfromthefactthatifa∈ A∩kerE then
AaA = E(AaA) = {0}, hence a = 0. Relation (10) is also straightforward,
since θ(B ) = B and E ◦θ = E . Indeed, for each x∈ B , we have
1 2 2 1 1
E (B θ(x)B ) = E (θ(B )θ(x)θ(B )) = E (θ(B xB )) = E (B xB ),
2 2 2 2 1 1 2 1 1 1 1 1
from which (10) follows.
To see that α(kerE) ⊆ kerE, choose k ∈ kerE. Since B is spanned by all
finiteproductsofelementsαn(a),a ∈ A,n =0,1,...,itsufficestoshowthat
E(yα(k)x) = 0 for all y ∈ B and all x of the form x= αm1(a )···αmk(a ).
1 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) ≤ kyk2E(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 > 0 for all i, then x = α(x )
i 0
for some x ∈ B, and using E◦α = ϕ◦E one has
0
∗ ∗ ∗ ∗ ∗
E(x α(k k)x) = E(α(x k kx )) =ϕ(E(x k kx )) = 0.
0 0 0 0
For the remaining case where some m = 0, notice that x must have one
i
of the forms x = a ∈ A (when all m are 0), or x = ax with x ∈ B
i 0 0
(when m = 0 and some other m is positive), or α(x )ax with a ∈ A,
1 j 1 2
x1,x2 ∈ B (when m1 > 0,...,mr−1 > 0 and mr = 0), and in each case
∗ ∗
E(x α(k k)x) = 0. If x = α(x )ax , for example, then we make use of
1 2
Formula (6) to write
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
E(x α(k k)x) = E(x a α(x k kx )ax )= E(x ) a E(α(x k kx )aE(x ).
2 1 1 2 2 1 1 2