Table Of ContentCOMPLETELY POSITIVE DEFINITE FUNCTIONS AND
BOCHNER’S THEOREM FOR LOCALLY COMPACT QUANTUM
GROUPS
2
1
0
MATTHEWDAWSANDPEKKASALMI
2
t
c Abstract. We prove two versions of Bochner’s theorem for locally compact
O quantum groups. First, every completely positive definite “function” on a
locally compact quantum group G arises as a transform of a positive func-
8 tionalontheuniversalC*-algebraC0u(Gˆ)ofthedualquantumgroup. Second,
1 when G is coamenable, complete positive definiteness may be replaced with
the weaker notion of positive definiteness, which models the classical notion.
] Acounterexampleisgiventoshowthatthelatterresultisnottrueingeneral.
A
To prove these results, we show two auxilary results of independent interest:
F products arelinearlydenseinL1(G),andwhenGiscoamenable, theBanach
♯
. ∗-algebraL1(G)hasacontractivebounded approximateidentity.
h ♯
t Keywords: Quantum group,positivedefinitefunction,Bochner’sTherorem.
a
m 2010MathematicsSubjectClassification. Primary: 20G42,43A35,Secondary:
22D25, 43A30,46L89.
[
1
v
1
3 1. Introduction
2
5 Bochner’s theorem (as generalised by Weil) tells us that any positive definite
. function on a locally compact abelian group G is the Fourier–Stieltjes transform
0
1 of a positive measure on the dual group Gˆ. In non-abelian harmonic analysis, we
2 can replace the algebra C (Gˆ) by the group C∗-algebra C∗(G), and hence replace
0
1 positive measures on Gˆ by positive functionals on C∗(G). Viewing C∗(G)∗ as
:
v B(G),theFourier–Stieltjesalgebra,Bochner’stheoremessentiallysaysthatpositive
i definitive functions are precisely the positive elements of B(G) (this viewpoint is
X
taken in [14, D´efinition 2.2]).
r
a For a locally compact quantum group G, we replace functions on groups by ele-
mentsofvonNeumann(orC∗-)algebras,whichcomeequippedwithextrastructure
reminiscentofanalgebrawhichreallyarisesfromagroup. GivenG,wecanformthe
universaldualalgebraCu(Gˆ),whichgeneralisesthepassagefromGtothefullgroup
0
C∗-algebraC∗(G) (see the next section for further details on Cu(Gˆ) and so forth).
0
Letting M(C (G) Cu(Gˆ))bethemaximalunitarycorepresentationofG,we
W ∈ 0 ⊗ 0
haveanalgebrahomomorphismCu(Gˆ)∗ M(C (G)) L∞(G);µˆ (id µˆ)( ∗).
0 → 0 ⊆ 7→ ⊗ W
In the commutative case, the image is precisely the Fourier–Stieltjes algebra (we
remark that it is slightly a matter of conventionif one uses or ∗ here). Moti-
W W
vated by this, there are perhaps two obvious notions for what a “positive definite”
element of L∞(G) should be; here we introduce some of our own terminology:
(1) A positive definite function is x L∞(G) with x∗,ω⋆ω♯ 0 for all
ω L1(G). ∈ h i ≥
∈ ♯
1
2 MATTHEWDAWSANDPEKKASALMI
(2) AFourier–Stieltjes transform of a positive measure isx L∞(G)suchthat
∈
there exists µˆ Cu(Gˆ)∗ with x=(id µˆ)( ∗).
∈ 0 + ⊗ W
It seems that definition (2) is a better fit with the current literature, although the
term “positive definite function” is not commonly used in this context (for exam-
ples where positive functionals in µˆ Cu(Gˆ)∗, or their transforms, are used in
∈ 0 +
place of positive definite functions (from the classical case), see [17] which stud-
ies Markov operators, [4, Section 4] and [3] which study various approximation
properties for von Neumann algebras over quantum groups, or [24] and [25] which
study property (T)for quantumgroups;the latter reference actually uses the term
“positive definite function” in an offhand way). Definition (1) is the most natural
as it directly generalises the notion of a positive definite function on L1(G), see
for example [11, Theorem 13.4.5]. This definition is rather briefly studied for Kac
algebras in [13, Section 1.3]; however, it is mainly (2), in various guises, which is
usedin[13]. Indeed,weshowinExample17belowthateveninthecocommutative
case,definition(1)isproblematicwithoutsomesortofamenabilityassumption–to
be precise, that G is coamenable. Even when G is coamenable, we are required to
deal with the unbounded antipode S, and our techniques are necessarily different
from those used for Kac algebras. We remark that it is easy to see that always
(2)= (1), see Lemma 1 below.
⇒
Definition (2) applied in the cocommutative case suggests that a “positive defi-
nite”elementofVN(G)shouldcomefromapositivemeasureinM(G)=C (G)∗ =
0
ML1(G) the multiplier algebra of L1(G). On the dual side, De Canni`ere and
Haagerupshowedin [10] that completely positive multipliers of A(G) coincide with
positive definite functions on G. This suggests the following notions:
(3) Acompletely positive multiplier isx L∞(G)suchthatthereexistsacom-
∈
pletely positive left multiplier L : L1(Gˆ) L1(Gˆ) with xλˆ(ωˆ)=λˆ(L (ωˆ))
x x
→
for every ωˆ L1(Gˆ). Here λˆ denotes the map ωˆ (id ωˆ)(W∗) =
∈ 7→ ⊗
(ωˆ id)(Wˆ) where W M(C (G) C (Gˆ)) is the left multiplicative uni-
0 0
tary⊗of G. ∈ ⊗
(4) A completely positive definite function is x L∞(G) such that there exists
a normal completely positive map Φ: (L2(∈G)) (L2(G)) with
B →B
x∗,ω ⋆ω♯ =(Φ(θ )β α)
h ξ,α η,βi ξ,η |
foreveryξ,η D(P1/2)andα,β D(P−1/2). HereP isadenselydefined,
positive, injec∈tive operator on L2(∈G) implementing the scaling group of G.
The first named author showed in [7] that (3) and (2) are equivalent notions, and
that they imply (4). We note that while (2) is the notion mostly adopted in the
literature (see above), it is actually the map L (or its adjoint) which is of interest
x
(the point being that the implication (2) = (3) is very easy to establish). Let
us motivate (4) a little more. The unbound⇒ed involution ♯ on L1(G) is given by
ω♯ =ω∗ S,wherethis is bounded. Normalcompletely positivemapson (L2(G))
biject wi◦th the positive part of the extended (or weak∗) Haagerup tensorBproduct
eh
(L2(G)) (L2(G)) (see [2, 12]), where a b is associated to Φ with
B ⊗B ⊗
Φ(θ)=aθb (Φ(θ )β α)= a,ω b,ω .
ξ,η ξ,α β,η
⇔ | h ih i
COMPLETELY POSITIVE DEFINITE FUNCTIONS AND BOCHNER’S THEOREM 3
eh
Hence(4)isequivalenttotheexistenceofapositiveu (L2(G)) (L2(G))with
∈B ⊗B
∗ ∗ ∗
∆(x ),ω ω S = u,ω ω .
h 1⊗ 2 ◦ i h 1⊗ 2i
Hence, informally, this is equivalent to (id S)∆(x∗) being a positive member of
⊗
(L2(G))eh (L2(G)). In the commutative case, x∗ = F L∞(G) say, and this
sBays that t⊗heBfunction (s,t) F(st−1) is, in some sense, a∈“positive kernel”, i.e.
that F is positive definite. So7→(4) saysthat x∗ defines a non-commutative,positive
definite kernel.
Weremarkthatasthe“inverse”operatoronG(theantipodeS)andtheadjoint
on L∞(G) do not commute, we have to be a little careful about using x or x∗ in∗
the above definitions.
The principle results of this paper are:
For any G, we have that (4) is equivalent to (3) and hence equivalent to
•
(2).
When G is coamenable (as is true in the commutative case!) all four con-
•
ditions are equivalent.
Both these results may be interpreted as versions of Bochner’s theorem for locally
compactquantumgroups. Whengivenaconditionlike(1)the obviousthing totry
is a GNS construction,in this caseapplied to the -algebraL1(G). In general,this
∗ ♯
algebra does not have an approximate identity, so we first show in Section 3 that
products are always linearly dense in L1(G), which enables a suitable GNS con-
♯
struction. InSection4weapplythis,togetherwithtechniquessimilartothoseused
in[7]toshowthat(4)= (3). OnecantakeasadefinitionthatGiscoamenableif
and only if L1(G) has a⇒bounded approximate identity. In Section 5 we show that
in this case, alsoL1(G) has a (contractive)approximateidentity. Indeed, we prove
♯
a slightly more general statement, adapting ideas of J. Kustermans, A. Van Daele
and J. Verding from [19] (we wish to approximate the counit ǫ, which is invariant
for the scaling group, and it is this invariance which is key; the argument in [19] is
for multiplier algebras of C∗-algebras and modular automorphism groups, and to
our mind, works because the unit of M(A) is invariant for the modular automor-
phismgroup). We suspectthatthe ideasofSections 3and5willprovetobe useful
in other contexts. In Section 6 we apply this to condition (1). In the final section
we consider n-positive multipliers.
2. Preliminaries
Throughout the paper G denotes a locally compact quantum group [21, 22,
23]. Its comultiplication ∆ is implemented by the left multiplicative unitary W
(L2(G) L2(G)): ∈
B ⊗
∆(x)=W∗(1 x)W (x L∞(G)).
⊗ ∈
The reduced C*-algebraC (G) is the norm closure of
0
(id ω)W : ω (L2(G))∗ .
{ ⊗ ∈B }
On the other hand, the norm closure of
(ω id)W : ω (L2(G))∗
{ ⊗ ∈B }
gives the reduced C*-algebra C (Gˆ) of the dual quantum group Gˆ. The left mul-
0
tiplicative unitary of the dual quantum group is just Wˆ = σW∗σ where σ is the
4 MATTHEWDAWSANDPEKKASALMI
flip map on L2(G) L2(G). The associated von Neumann algebras L∞(G) and
⊗
L∞(Gˆ) are the weak∗-closures of the respective C*-algebras C (G) and C (Gˆ).
0 0
The predual L1(G) of L∞(G) is a Banach algebra under the convolution product
ω⋆τ = (ω τ)∆. Given ξ,η L2(G), let ω L1(G) be the normal functional
ξ,η
x (xξ η⊗). As L∞(G) is in s∈tandard position o∈n L2(G), every member of L1(G)
7→ |
arises in this way.
The scaling group (τ ) of G is implemented by a positive, injective, densely de-
t
fined operator P on L2(G): we have τ (x)= PitxP−it. Then the antipode S of G
t
hasa polar decompositionS =Rτ−i/2, where R is the unitary antipode. Inpartic-
ular ∆R=(R R)σ∆, where σ denotes the flip map, now on L∞(G) L∞(G).
We follow [2⊗0, Section 3] to define the -algebra L1(G). Recall that⊗ω L1(G)
∗ ♯ ∈
is a member of L1(G) if and only if there exists ω♯ L1(G) such that
♯ ∈
x,ω♯ = S(x)∗,ω (x D(S))
h i h i ∈
where D(S) denotes the domain of S. Then L1(G) is a dense subalgebra ofL1(G).
♯
ThenaturalnormofL1(G)is ω =max( ω , ω♯ ),andwith♯astheinvolution,
♯ k k♯ k k k k
L1(G) is a Banach -algebra. For ω L1(G), let ω∗ L1(G) be the functional
♯ ∗ ∈ ∈
x,ω∗ = x∗,ω . Thus ω L1(G) if and only if w∗ S is bounded. Notice that
has ∆ iis a h-homiomorphism∈, th♯e map ω ω∗ is an an◦ti-linear homomorphism on
L1(G); wh∗ile ω ω♯ is an anti-linear an7→ti-homomorphism on L1(G).
7→ ♯
The universalC*-algebraCu(Gˆ) associated to Gˆ is the universal C*-completion
0
of L1(G) (see [20] for details). The natural map (i.e. the universal representa-
♯
tion) λ : L1(G) Cu(Gˆ) is implemented as λ (ω) = (ω id)( ) where
u ♯ → 0 u ⊗ W W ∈
M(C (G) Cu(Gˆ))isthemaximalunitarycorepresentationofG(whichisdenoted
0 ⊗ 0
by ˆ in [20, Proposition 4.2]).
V
Lemma 1. Let x=(id µˆ)( ∗) for some µˆ Cu(Gˆ)∗. Then x∗,ω⋆ω♯ 0 for
⊗ W ∈ 0 + h i≥
every ω L1(G).
∈ ♯
Proof. Simply note that
x∗,ω⋆ω♯ = (∆ id) ,ω ω♯ µˆ = ,ω ω♯ µˆ
13 23
h i h ⊗ W ⊗ ⊗ i hW W ⊗ ⊗ i
= µˆ,λ (ω)λ (ω♯) = µˆ,λ (ω)λ (ω)∗ 0,
u u u u
h i h i≥
as required. (cid:3)
Similarly to [18], we say that x M(C (G)) is a left multiplier of L1(Gˆ) if
0
∈
xλˆ(ωˆ) λˆ L1(Gˆ) whenever ωˆ L1(Gˆ),
∈ ∈
(cid:0) (cid:1)
where
λˆ: C (Gˆ)∗ M(C (G)), λˆ(µˆ)=(µˆ id)(Wˆ)=(id µˆ)(W∗).
0 0
→ ⊗ ⊗
In this case we can define L : L1(Gˆ) L1(Gˆ) by
x
→
λˆ(L (ωˆ))=xλˆ(ωˆ)
x
becauseλˆ isinjective. WeseeimmediatelythatL isaleftmultiplier(oftentermed
x
a“leftcentraliser”intheliterature)intheusualsense,thatis,L (ωˆ⋆τˆ)=L (ωˆ)⋆τˆ
x x
for everyωˆ,τˆ L1(Gˆ). The following lemma is shownfor Kac algebrasin [18], but
∈
since we need the (short) argument once more, we include a proof.
COMPLETELY POSITIVE DEFINITE FUNCTIONS AND BOCHNER’S THEOREM 5
Lemma 2. Let x M(C (G)) be a left multiplier of L1(Gˆ). Then L : L1(Gˆ)
0 x
∈ →
L1(Gˆ) is bounded.
Proof. Weapplythe closedgraphtheorem. Supposethatωˆ ωˆ andL (ωˆ ) τˆ
n x n
→ →
in L1(Gˆ). Then
λˆ(L (ωˆ)) λˆ(τˆ) xλˆ(ωˆ) xλˆ(ωˆ ) + λˆ(L (ωˆ )) λˆ(τˆ)
x n x n
k − k≤k − k k − k
x ωˆ ωˆ + L (ωˆ ) τˆ 0
n x n
≤k kk − k k − k→
asn . Sinceλˆisinjective,wehaveL (ωˆ)=τˆandbytheclosedgraphtheorem,
x
L is→bo∞unded. (cid:3)
x
We say that x M(C (G)) is an n-positive multiplier if it is a left multiplier
0
∈
of L1(G) and the map L∗: L∞(Gˆ) L∞(Gˆ) is n-positive. We shall consider n-
x →
positive multipliers more carefully in Section 7, but the main interest of the paper
shall be the completely positive multipliers: that is, x M(C (G)) that are n-
0
positive multipliers for every n N. When x is a comp∈letely positive multiplier,
L∗ extends to a normal, comple∈tely positive map Φ: (L2(G)) (L2(G)) (see
x B → B
[16, Proposition 4.3] or [7, Proposition 3.3]). Moreover,by [7, Proposition 6.1],
x∗,ω ⋆ω♯ =(Φ(θ )β α)
h ξ,α η,βi ξ,η |
for every ξ,η D(P1/2) and α,β D(P−1/2), so x is completely positive definite.
∈ ∈
We shall prove the converse in section 4, but first we need a bit of groundwork.
The following is similar to known results about cores for analytic generators
(compare [29, Theorem X.49] for example) but we give the short proof for com-
pleteness. Let us just remark that as S = Rτ−i/2 and τt(x) = PitxP−it for all t,
the functional ω is in L1(G) whenever ξ D(P1/2) and α D(P−1/2), and in
ξ,α ♯ ∈ ∈
this case ωξ♯,α =R∗(ωP−1/2α,P1/2ξ)=ωJˆP1/2ξ,JˆP−1/2α; see [7, Section 6].
Lemma 3. The set
D = ω : ξ D(P1/2),α D(P−1/2)
ξ,α
{ ∈ ∈ }
is dense in L1(G) with respect to its natural norm (i.e. D is a core for ω ω♯).
♯ 7→
Proof. For ω L1(G), r >0, define
∈
∞
ω(r)= r e−r2t2ω τ dt.
t
√π Z−∞ ◦
See also Section 5 below. Since the modular group (τ ) is implemented by P, it
t
follows that D is invariant under (τ ). On the other hand, since R commutes with
t
(τt) and S = Rτ−i/2, we have (ω ◦τt)♯ = ω♯ ◦ τt for every ω ∈ L1♯(G), and so
t ω τ iscontinuouswithrespecttothenormofL1(G). Consequently,ifω D,
7→ ◦ t ♯ ∈
then ω(r) is in the L1(G)-closure of D for every r >0.
♯
Given ω L1(G), there exists (ω ) D such that ω ω in L1(G), because
∈ ♯ n ⊆ n →
L∞(G)isinstandardformonL2(G)andthe domainsofP1/2 andP−1/2 aredense
in L2(G). By the beginning of the proof, ω (r) is in the L1(G)-closure of D. A
n ♯
simple calculation shows that
ω(r) ω (r) er2/4 ω ω ,
n ♯ n
k − k ≤ k − k
6 MATTHEWDAWSANDPEKKASALMI
and so ω(r) is in the L1(G)-closure of D. As r , we have ω(r) ω and
♯ → ∞ →
ω(r)♯ = ω♯(r) ω♯ (since ω L1(G)). Therefore ω is in the L1(G)-closure of D,
as claimed. → ∈ ♯ ♯ (cid:3)
3. Density of products in L1(G)
♯
The main result of this section is that, in analogy to L1(G), the convolution
products are linearly dense in L1(G) with respect to its natural norm. We shall
♯
prove this result based on two (closely related) lemmas, the first of which is from
[5, Proposition A.1].
Lemma 4. Let x,y L∞(G) satisfy y,ω♯ = x∗,ω∗ for all ω L1(G). Then
∈ h i h i ∈ ♯
y D(S) and S(y)=x∗.
∈
Lemma 5. Let x,y L∞(G) satisfy y,(ω ⋆ω )♯ = x∗,ω∗⋆ω∗ for all ω ,ω
L1(G). Then y D(∈S) with S(y)=xh∗. 1 2 i h 1 2i 1 2 ∈
♯ ∈
Proof. For n N define the smear
∈
∞
y(n)= n e−n2t2τ (y) dt
t
√π Z−∞
where (τ ) is the scaling group. Define x(n) similarly using x∗.
t
As R commutes with (τt) and S =Rτ−i/2, it follows that ω♯ τt =(ω τt)♯ for
◦ ◦
ω L1(G). Using that ∆ τ =(τ τ ) ∆ we find that for ω ,ω L1(G)
∈ ♯ ◦ t t⊗ t ◦ 1 2 ∈ ♯
∞
∆(y(n)),ω♯ ω♯ = n e−n2t2 ∆(τ (y)),ω♯ ω♯ dt
h 2⊗ 1i √π Z−∞ h t 2⊗ 1i
∞
= n e−n2t2 ∆(y),(ω τ )♯ (ω τ )♯ dt
2 t 1 t
√π Z−∞ h ◦ ⊗ ◦ i
∞
= n e−n2t2 ∆(x∗),(ω τ )∗ (ω τ )∗ dt
1 t 2 t
√π Z−∞ h ◦ ⊗ ◦ i
∗ ∗
= ∆(x(n)),ω ω .
h 1 ⊗ 2i
Asy(n) D(S)thevonNeumannalgebraicversionof[23,Lemma5.25]showsthat
∈
∆(y(n)),ω♯ ω♯ = ∆(S(y(n))),ω∗ ω∗ .
h 2⊗ 1i h 1 ⊗ 2i
Thus ∆(S(y(n)))=∆(x(n)), andas ∆is injective, S(y(n))=x(n). Nowy(n) y
in the σ-weak topology, and x(n) x∗. As S is a σ-weakly closed operato→r, it
follows that y D(S) with S(y)=→x∗, as required. (cid:3)
∈
Theorem6. LetGbealocallycompactquantumgroup. Then ω⋆τ : ω,τ L1(G)
{ ∈ ♯ }
is linearly dense in L1(G) in its natural norm.
♯
Proof. For a Banach space E, let E be the conjugate space to E. For x E let
∈
x E be the image of x, so x+y = x+y and tx = tx for x,y E,t C. We
ide∈ntify (E)∗ with E∗ via µ,x = µ,x . ∈ ∈
h i h i
Then the map
L1♯(G)→L1(G)⊕∞L1(G), ω 7→(ω,ω♯)
COMPLETELY POSITIVE DEFINITE FUNCTIONS AND BOCHNER’S THEOREM 7
is a linear isometry. Thus the adjoint L∞(G) L∞(G) L1(G)∗ is a quotient
⊕1 → ♯
map. So anymember ofL1(G)∗ is induced by a pair(x,y)with x,y L∞(G), and
♯ ∈
the dual pairing is
(x,y),ω = x,ω + y,ω♯ = x,ω + y,ω♯ .
h i h i h i h i h i
Firstly, (x,y)=0 if andonly if x∗,ω∗ = y,ω♯ for all ω L1(G) if and only
if, by Lemma 4, y D(S) with S(hy−)= x∗i. h i ∈ ♯
Nowlet(x,y)an∈nihilateallelements−ofthe formω⋆τ, withω,τ L1(G). Then
∈ ♯
0= x,ω⋆τ + y,τ♯⋆ω♯ = x∗,ω∗⋆τ∗ = y,τ♯⋆ω♯ .
h i h i ⇒ h− i h i
By Lemma 5, y D(S) with S(y) = x∗. That is, (x,y) = 0. So by the Hahn–
Banach theorem,∈the result follows. − (cid:3)
4. Completely positive definite functions
The definition of completely positive definite functions on a locally compact
quantumgroupGwasproposedbythefirstnamedauthorin[7],andinthissection
we show that, as conjectured in [7], such elements are precisely the completely
positive multipliers. This result may be viewed as a version of Bochner’s theorem
because the completely positive multipliers are known by [7] to be of the form
(id µˆ)( ∗), with µˆ Cu(Gˆ)∗.
⊗ W ∈ 0 +
We begin with a preliminary result, also of independent interest.
Proposition 7. Let x L∞(G) be completely positive definite. Then x∗ D(S)
and S(x∗)=x. ∈ ∈
Proof. For every ξ,η D(P1/2) and α,β D(P−1/2)
∈ ∈
x∗,ω ⋆ω♯ = Φ(θ ),ω = Φ(θ∗ ),ω∗ = Φ(θ ),ω
h ξ,α η,βi h ξ,η β,αi h ξ,η β,αi h η,ξ α,βi
= x∗,ω ⋆ω♯ = x,ω∗ ⋆ω♯∗ = x,(ω ⋆ω♯ )♯∗ .
h η,β ξ,αi h η,β ξ,αi h ξ,α η,β i
It follows from Theorem 6 and Lemma 3 that
x∗,ω = x,ω♯∗
h i h i
for every ω L1(G). Then it follows from Lemma 4 that x∗ D(S) and S(x∗) =
x. ∈ ♯ ∈ (cid:3)
The following lemma shows that a GNS-type construction works for positive
definite functions. Compare with [11, Proposition2.4.4],but note that L1(G) does
♯
notnecessarilyhavea boundedapproximateidentity, the lackofwhichis remedied
by the density of products in L1(G) (Theorem 6).
♯
Lemma 8. Let x L∞(G) be positive definite. Then
∈
(ω τ)= x∗,τ♯⋆ω
| h i
defines a pre-inner-product on L1(G); let Λ: L1(G) H be the associated map to
♯ ♯ →
the Hilbert space H obtained by completion. Then,
π(ω)Λ(τ)=Λ(ω⋆τ)
defines a non-degenerate -representation π of L1(G) on H.
∗ ♯
8 MATTHEWDAWSANDPEKKASALMI
Proof. Such GNS-type results are usually stated for algebras with an approximate
identity(seeforexample[6,Section3.1]or[11,Section2.4])soforcompleteness,we
give the details in this slightly more general setting. When x L∞(G) is positive
∈
definite,
(ω τ)= x∗,τ♯⋆ω .
| h i
defines a positive sesquilinear form on L1(G). As in the statement, let H be the
♯
Hilbert space completion of L1(G)/ , where denotes the associated null space,
♯ N N
and let Λ: L1(G) H be the map taking ω L1(G) to the image of ω+ in H.
♯ → ∈ ♯ N
We are left to show that
π(ω)Λ(τ)=Λ(ω⋆τ)
definesanon-degenerate -representationofL1(G)onH. Ifwecanshowthatπ(ω)
∗ ♯
is well-defined and bounded, then it follows easily that π is a -homomorphism.
Let r denote the spectral radius on L1(G). By [6, Corollary∗3.1.6],
♯
π(ω)Λ(τ) 2 = x∗,τ♯⋆ω♯⋆ω⋆τ r(ω♯⋆ω) x∗,τ♯⋆τ =r(ω♯⋆ω) Λ(τ) 2.
k k h i≤ h i k k
This shows that π(ω) maps to and hence is well-defined. Moreover, we see
N N
that π(ω) defines a bounded operator on H.
Finally, since L1(G) is the closed linear span of L1(G)⋆L1(G) by Theorem 6
♯ ♯ ♯
and Λ is continuous, the -representation π is non-degenerate. (cid:3)
∗
Theorem 9. An element x L∞(G) is completely positive definiteif and only if it
∈
is a completely positive multiplier. In particular, every completely positive definite
x L∞(G) is in M(C (G)).
0
∈
Proof. As already noted, the “if” part is proved in [7], so we let x L∞(G)
be completely positive definite. Let Φ: (L2(G)) (L2(G)) be the a∈ssociated
B → B
completely positive map such that
x∗,ω ⋆ω♯ =(Φ(θ )β α)
h ξ,α η,βi ξ,η |
whenever ξ,η D(P1/2) and α,β D(P−1/2). Then Φ has a Stinespring dilation
∈ ∈
of the form
Φ(θ)=V∗(θ 1)V (θ (L2(G))),
0
⊗ ∈B
whereV : L2(G) L2(G) K is a boundedmap forsome Hilbert space K (see [7,
→ ⊗
Section 5] for details). Letting (e ) be an orthonormal basis of K, we can define a
i
family (a ) in (L2(G)) with a∗a < such that
i B i i i ∞
P
Vξ = a ξ e
i i
⊗
Xi
and hence
∗
Φ(θ)= a θa .
i i
Xi
We may also take the Stinespring dilation to be minimal so that
(θ id)Vξ : ξ L2(G),θ (L2(G))
0
{ ⊗ ∈ ∈B }
is linearly dense in L2(G) K. This is equivalent to vectors of the form
⊗
ai,ω η ei (ω (L2(G))∗,η L2(G))
h i ⊗ ∈B ∈
Xi
being linearly dense; equivalently that vectors of the form a ,ω e are dense in
K as ω (L2(G))∗ varies. Pih i i i
∈B
COMPLETELY POSITIVE DEFINITE FUNCTIONS AND BOCHNER’S THEOREM 9
Let (Λ,π,H) be the GNS construction for x from Lemma 8. Then, for ξ,η
D(P1/2) and α,β D(P−1/2), ∈
∈
Λ(ω♯ ) Λ(ω♯ ) = x∗,ω ⋆ω♯ =(Φ(θ )α β)= (a (α) ξ)(a (β) η).
ξ,α η,β H h η,β ξ,αi η,ξ | i | i |
(cid:0) (cid:12) (cid:1) Xi
(cid:12)
Letq : (L2(G))∗ L1(G) be the quotient map. Since the functionals of the form
B →
ω♯ are dense in L1(G) by Lemma 3, it follows that there is an isometry
ξ,α ♯
v: H →K; Λ(q(ω)♯)7→ hai,ω∗iei (ω ∈B(L2(G))∗,q(ω)∈L1♯(G))
Xi
(note that a ,ω∗ e 2 a ,ω∗ 2 ω 2 a∗a ). As we have a
k ih i i ik ≤ i|h i i| ≤ k k k i i ik
minimal StinePspring dilation, vPhas dense range and is Phence unitary. A corollary
of v even being well-defined is that for each i and each ω (L2(G))∗, the value
of a ,ω depends only on the value q(ω). Hence a L∞(G∈)Bfor each i.
i i
hConsiider now the non-degenerate -representati∈on of L1(G) on K given by
vπ()v∗. By Kustermans [20, Corollary∗4.3], there is an asso♯ciated unitary corep-
rese·ntation U of G; so U L∞(G) (K) and vπ(ω)v∗ = (ω id)(U). As U is a
unitarycorepresentation,w∈eknowth⊗aBt(ω♯ id)(U)=(ω id)(U⊗)∗ =(ω∗ id)(U∗).
If ω∗ L1(G), it follows that (ω id)(U∗)⊗=vπ(ω∗♯)v∗,⊗and thus ⊗
∈ ♯ ⊗
∗ ∗
U ξ a ,ω e α e = (ω id)(U ) a ,ω e e
i i j ξ,α i i j
(cid:16) (cid:0) ⊗Xi h i (cid:1)(cid:12)(cid:12) ⊗ (cid:17) (cid:16) ⊗ Xi h i (cid:12)(cid:12) (cid:17)
(cid:12) (cid:12)
= vπ(ω♯ )v∗ a ,ω e e = vπ(ω♯ )Λ(ω∗♯) e
(cid:16) α,ξ Xi h i i i (cid:12)(cid:12) j(cid:17) (cid:16) α,ξ (cid:12)(cid:12) j(cid:17)
= vΛ(ω♯ ⋆ω∗♯) e = a(cid:12) ,(ω∗⋆ω )∗ = a ,ω⋆(cid:12)ω
α,ξ j h j α,ξ i h j ξ,αi
(cid:0) (cid:12) (cid:1)
= (ω id)∆(a(cid:12) )ξ e α e
i i j
(cid:16)Xi ⊗ ⊗ (cid:12)(cid:12) ⊗ (cid:17)
(cid:12)
whenever ξ D(P1/2), α D(P−1/2). As an aside, we note that this is precisely
thewayinw∈hichU∗ isdefi∈nedin[7,Proposition5.2]. Thus,perhapsasexpected,if
x comes froma completely positive multiplier, then the two approachesto forming
representations agree.
Since U is unitary, we have, for every ξ,η D(P1/2) and ω ,ω L1(G)∗,
∈ 1 2 ∈ ♯
∗ ∗
(ξ η) a ,ω a ,ω = U ξ a ,ω e U η a ,ω e
i 1 i 2 i 1 i i 2 i
| Xi h ih i (cid:16) (cid:0) ⊗Xi h i (cid:1)(cid:12)(cid:12) (cid:0) ⊗Xi h i (cid:1)(cid:17)
(cid:12)
= (ω id)∆(a )ξ e (ω id)∆(a )η e
1 i i 2 i i
(cid:16)Xi ⊗ ⊗ (cid:12)(cid:12)Xi ⊗ ⊗ (cid:17)
∗ (cid:12)
= (ω id)∆(a ) (ω id)∆(a )ξ η .
2 i 1 i
⊗ ⊗
Xi (cid:0) (cid:12) (cid:1)
(cid:12)
It follows that
∗ ∗ ∗ ∗
a a 1,ω ω ω = ∆(a ) ∆(a ) ,ω ω ω .
h i ⊗ i⊗ 2 ⊗ 1⊗ ξ,ηi h i 13 i 23 2 ⊗ 1⊗ ξ,ηi
Xi Xi
As this holds for a dense collection of ω ,ω ,ξ,η it follows that
1 2
∗ ∗
a a 1= ∆(a ) ∆(a )
i ⊗ i⊗ i 13 i 23
Xi Xi
10 MATTHEWDAWSANDPEKKASALMI
(recallthat a∗a < so the sums on both sides do convergeσ-weakly.). Then,
for θξ,η 0P(Li2(iG)i) an∞d α,β L2(G),
∈B ∈
∗ ∗
(Φ(θ )β α)1= (a θ a β α)1= (a ξ α)(a β η)1
ξ,η | i ξ,η i | i | i |
Xi Xi
∗
= (ω ω id)(a a 1)
ξ,α⊗ β,η⊗ i ⊗ i⊗
Xi
∗
= (ω ω id)∆(a ) ∆(a )
ξ,α⊗ β,η⊗ i 13 i 23
Xi
∗
= (ω id)∆(a )(ω id)∆(a )
ξ,α⊗ i β,η⊗ i
Xi
∗
= (ω id) ∆(a )(θ 1)∆(a ) .
β,α⊗ i ξ,η⊗ i
Xi (cid:0) (cid:1)
It follows that
Φ(θ) 1= ∆(a∗)(θ 1)∆(a ) (θ (L2(G))).
⊗ i ⊗ i ∈B0
Xi
By normality, and using that ∆()=W∗(1 )W, we see that for y (L2(G)),
· ⊗· ∈B
∗ ∗ ∗
Φ(y) 1= W (1 a )W(y 1)W (1 a )W
⊗ ⊗ i ⊗ ⊗ i
Xi
and hence
1 Φ(y)= Wˆ(a∗ 1)Wˆ∗(1 y)Wˆ(a 1)Wˆ∗
⊗ i ⊗ ⊗ i⊗
(1) Xi
=Wˆ (Φ id)(Wˆ∗(1 y)Wˆ) Wˆ∗.
⊗ ⊗
(cid:0) (cid:1)
In particular, for xˆ L∞(Gˆ),
∈
Wˆ∗(1 Φ(xˆ))Wˆ =(Φ id)∆ˆ(xˆ).
⊗ ⊗
Now,theleft-hand-sideisamemberofL∞(Gˆ) (L2(G)),andtheright-hand-side
⊗B
isamemberof (L2(G)) L∞(Gˆ),andsobothsidesarereallyinL∞(Gˆ) L∞(Gˆ)
B ⊗ ⊗
(by taking bicommutants for example). Then
(Φ id) ∆ˆ(xˆ)(1 yˆ) = (Φ id)∆ˆ(xˆ) (1 yˆ) L∞(Gˆ) L∞(Gˆ),
⊗ ⊗ ⊗ ⊗ ∈ ⊗
(cid:0) (cid:1) (cid:0) (cid:1)
and so, as ∆ˆ(xˆ)(1 yˆ) : xˆ,yˆ L∞(Gˆ) is a σ-weakly, linearly dense subset of
{ ⊗ ∈ }
L∞(Gˆ) L∞(Gˆ), it follows that Φ maps L∞(Gˆ) to L∞(Gˆ). (We remark that
⊗
the fact that ∆ˆ(xˆ)(1 yˆ) : xˆ,yˆ L∞(Gˆ) is linearly σ-weakly dense is shown
{ ⊗ ∈ }
directly in the remark after [32, Proposition 1.21]. One can prove this by noting
thatD(S)is σ-weaklydense,andthenusingthe vonNeumannalgebraicversionof
the characterisationof S given by [23, Corollary 5.34].)
Let L be the restriction of Φ to L∞(Gˆ). Then the calculation above shows that
∆ˆL=(L id)∆ˆ,
⊗
andsoListheadjointofacompletelypositiveleftmultiplieronL1(Gˆ). Comparing
(1) with [7, Proposition 3.3] we see that Φ coincides with the extension of L used
in [7, 16]. In particular, by [7, Propositions 3.2] there is x M(C (G)) such that
0 0
∈
∗ ∗ ∗
(x 1)W =(id L)(W )=(id Φ)(W ).
0
⊗ ⊗ ⊗
