Cuntz-Pimsner Algebras and Twisted Tensor Products

0.33 MB·

by  Adam Morgan

CUNTZ-PIMSNER ALGEBRAS AND TWISTED
TENSOR PRODUCTS
6
1
ADAM MORGAN
0
2

n
a
J Abstract. Given two correspondences X and Y and a discrete

8 groupGwhichactsonX andcoactsonY,onecandefineatwisted
2 tensor product X ⊠Y which simultaneously generalizes ordinary

] tensor products and crossed products by group actions and coac-
A tions. Weshowthat,undersuitableconditions,theCuntz-Pimsner
O algebra of this product, OX⊠Y, is isomorphic to a “balanced”
. twisted tensor product OX ⊠TOY of the Cuntz-Pimsner algebras
h
of the original correspondences. We interpret this result in sev-
t
a eral contexts and connect it to existing results on Cuntz-Pimsner
m
algebras of crossed products and tensor products.
[

1
v 1. Introduction
6
2 In the author’s previous paper [Mor15], it was shown that, un-
8 der suitable conditions, if X and Y are C∗-correspondences over C∗-
7
algebras A and B, the Cuntz-Pimsner algebra O is isomorphic to
0 X⊗Y
1. a subalgebra OX ⊗T OY of OX ⊗ OY. In the present paper, we will
0 extend this result from ordinary tensor products to a certain class of
6 “twisted” tensor products.
1
Manyconstructionsinoperatoralgebrasmaybethoughtofas“twisted”
:
v
tensor products, for example: crossed products by actions or coac-
i
X tions of groups, Z -graded tensor products and so on. In [MRW14], a
2
r very general construction of a “twisted tensor product” is presented.
a
Their construction involves two quantum groups G = (S,∆ ) and
S
H = (T,∆ ), two coactions (A,G,δ ) and (B,H,δ ), and a bichar-
T A B
acter χ ∈ U(S ⊗ T). Given this information, they define a twisted
tensor product A⊠ B. They also show that if X and Y are correspon-
χ
dences over AbandbB with compatible coactions of G and H, there is
a natural way of defining a correspondence X ⊠ Y over A⊠ B. In
χ χ
this paper, we will work with a special case of this general construc-
tion which is general enough to be useful but simple enough to be very
tractable.
Date: January 29, 2016.
1
2 ADAMMORGAN
Specifically, we are interested in the case where, for some discrete
\
group G, S = c (G), T = C∗(G) and χ = WG ∈ U C∗(G)⊗C∗(G) is
0 r r r
the reduced bicharacter of C∗(G) viewed as a quantum group. In this
r (cid:0) (cid:1)
case, we may view the coaction of c (G) as an action of G on A (or
0
X), and we will be able to describe most of the algebraic properties of
A⊠ B and X ⊠ Y entirely in terms of elementary tensors.
χ χ
In this simplified setting, we will prove our main result: if JX⊠χY =
JX⊠χJY thenOX⊠χY ∼= OX⊠χOY (whereJX = φ−1 K(X) ∩ ker(φ) ⊥
is the Katsura ideal). We will then apply this result to some specific
(cid:0) (cid:1) (cid:0) (cid:1)
examples.
2. Preliminaries
2.1. Correspondences and Cuntz-Pimsner Algebras. For a gen-
eral reference on correspondences, we refer the reader to [Lan95]. For
Cuntz-Pimsner algebras, we recommend [Kat04], [Kat03] and the brief
overview in [Rae05]. We will briefly recall some of the basic facts here.
Suppose A is a C∗-algebra and X is a right A-module. We say that
X has an A-valued inner product if there is a map
X ×X ∋ (x,y) 7→ hx,yi ∈ A
A
which is A-linear in the second variable and satisfies the following
(1) hx,xi ≥ 0 for all x ∈ X with equality if and only if x = 0
A
(2) hx,yi∗ = hy,xi for all x,y ∈ X
A A
(3) hx,y ·ai = hx,yi a for all x,y ∈ X and a ∈ A.
A A
We can define the following norm on X:
1
kxkA := khx,xiAk2
If X is complete under the norm k·k defined above, X is called a
A
right Hilbert A-module. Note that if A = C then X is just a Hilbert
spaceandwecanthinkofHilbertmodulesasgeneralizedHilbertspaces
where the scalars are elements of some C∗-algebra A. Sometimes we
will write (X,A) or X if we wish to emphasize A.
A
Let A be a C∗-algebra and let X be a right Hilbert A-module. Sup-
pose T : X → X is an A-module homomorphism. If there is an A-
module homomorphism T∗ such that
∗
hT x,yi = hx,Tyi
A A
for all x,y ∈ X, then we call T adjointable and we refer to T∗ as
the adjoint of T. The set of all adjointable operators on X with the
operatornorm is a C∗-algebra. Wedenote this algebra by L(X). Given
x,y ∈ X we define the operator Θ as follows: Θ (z) = xhy,zi .
x,y x,y A
CUNTZ-PIMSNER ALGEBRAS AND TWISTED TENSOR PRODUCTS 3
The closed linear span of all such operators is a subalgebra of L(X)
whichwecallthegeneralized compact operators. ItisdenotedbyK(X).
Suppose X is a right Hilbert B-module. Suppose further that we
have a homomorphism φ : A → L(X) for some C∗-algebra A. This
is called a left action of A by adjointable operators. We call the triple
(A,X,B) a C∗-correspondence or simply a correspondence. For a ∈ A
and x ∈ X, we will write a · x for φ(a)(x). If A = B we call this a
correspondence over A (or B). We call the left-action injective if φ is
injective and non-degenerate if φ(A)X = X. If φ(A) ⊆ K(X), we say
that the left action is implemented by compacts. We will sometimes
write X to indicate that X in an A − B correspondence. Given
A B
an A − B correspondence X and an A − B correspondence Y, a
1 2 2 2
correspondence isomorphism is a triple (ϕ ,Φ,ϕ ) where Φ is a linear
A B
isomorphism Φ : X → Y and ϕ : A → A and ϕ : B → B are
A 1 2 B 1 2
isomorphisms of C∗-algebras such that the left and right actions and
the inner product are preserved by the maps:
Φ(ax) = ϕ (a)Φ(x)
A
Φ(xb) = Φ(x)ϕ (b)
B
Φ(x),Φ(x′) Y = ϕ x,x′iX
B B B
(cid:16) (cid:17)
(cid:10) (cid:11) (cid:10)
Where h·,·iY denotes the B-valued inner product on Y and h·,·iX de-
B B
notes the B-valued inner product on X. It will be convenient to intro-
duce the following definition:
Definition 2.1. Let X be an A − B correspondence. A generating
system for X is a triple (A0,X0,B0) where A0 ⊆ A, X0 ⊆ X and
B0 ⊆ B such that span(A0) = A, span(X0) = X, and span(B0) = B
and such that for all x ∈ X0 we have that ax,xb ∈ X0 for all a ∈ A0
and b ∈ B0. If A = B and A0 = B0 we will denote the generating
system by (X0,A0).
We will make frequent use of the following fact:
Lemma 2.2. Let X be an A −B correspondence and Y be a A −B
1 1 2 2
correspondence. Suppose that (A0,X0,B0) and (A0,Y0,B0) are gen-
1 1 2 2
erating sets for X and Y respectively. Let ϕ : A → A and ϕ :
A 1 2 B
B → B be isomorphisms. Suppose there is a bijection Φ : X0 → Y0,
1 2 0
which preserves the inner product, left and right actions, and scalar
4 ADAMMORGAN
multiplication. That is
Φ (x),Φ (x′) Y = ϕ hx,x′iX for all x,x′ ∈ X0
0 0 B B B
(cid:10) Φ (a(cid:11)x) = ϕ ((cid:16)a)Φ (x)(cid:17) for all x ∈ X0 and a ∈ A0
0 A 0
Φ (xb) = Φ (x)ϕ (b) for all x ∈ X0 and b ∈ B0
0 0 B
Φ (cx) = cΦ (x) for all x ∈ X0 and c ∈ C
0 0
Then Φ extends linearly and continuously to a correspondence isomor-
0
phism Φ : X → Y.
Proof. Let x ∈ span(X0), then x = c x for some x ∈ X0 and
i i i i
c ∈ C. We define Φ(x) = c Φ (x ). First, we must verify that
i i i 0 i
Φ is well defined on span(X ). SuppoPse n c x and m d x′ are
both equal to x ∈ X with x0P,x′ ∈ X0. Leit=1y i=i n c Φi=1(xi)iand
y′ = m d Φ (x′). Then i i P i=1Pi 0 i
i=1 i 0 i P
P
ky −y′k2 = y −y′,y −y′iB′
n m n m
(cid:10) ′ ′
= c Φ (x )− d Φ (x ), c Φ (x )− d Φ (x )
i 0 i i 0 i i 0 i i 0 i
B′
DXi=1 Xi=1 Xi=1 Xi=1 E
n,n m,m
′ ′
= cicjhΦ0(xi),Φ0(xj)iB′ + didjhΦ0(xi),Φ0(xj)iB′
i,j=1 i,j=1
X X
n,m m,n
′ ′
− cidjhΦ0(xi),Φ0(xj)iB′ − dicjhΦ0(xi),Φ0(xj)iB′
i,j=1 i,j=1
X X
n,n m,m
′ ′
= c c ϕ hx ,x i + d d ϕ hx ,x i
i j B i j B i j B i j B
i,j=1 i,j=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
n,m m,n
′ ′
− c d ϕ hx ,x i − d c ϕ hx ,x i
i j B i j B i j B i j B
i,j=1 i,j=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
n m n m
′ ′
= ϕ c x − d x , c x − d x
B i i i i i i i i
B
(cid:16)DXi=1 Xi=1 Xi=1 Xi=1 E (cid:17)
= ϕ hx−x,x−xi
B B
= 0
(cid:0) (cid:1)
where we have used the fact that ϕ is an isomorphism and thus linear.
CUNTZ-PIMSNER ALGEBRAS AND TWISTED TENSOR PRODUCTS 5
For x,x′ ∈ X0 we have
Y
′ Y ′ ′
Φ(x),Φ(x) = c Φ (x ), c Φ (x )
B i 0 i j 0 j
B
(cid:10) (cid:11) DXi Xj E
′ ′ Y
= c c Φ (x ),Φ (x )
i j 0 i 0 j B
i,j
X (cid:10) (cid:11)
= c c′ϕ hx ,x′iX
i j B i j B
Xi,j (cid:16) (cid:17)
X
′ ′
= ϕ c x , c x
B i i j j
B
(cid:16)DXi Xj E (cid:17)
= ϕ hx,x′iX
B B
(cid:16) (cid:17)
Therefore, Φ preserves the inner product on span(X0) and thus also
preserves the norm on span(X0) and so Φ is bounded and can be ex-
tended continuously to span(X0) = X. Hence, for any z ∈ X we can
approximate z ≈ c z with z ∈ X0. Thus for any a ∈ A0 we have
i i i i 0
Φ(a z) ≈ Φ a Pc z = c Φ(a z ) = ϕ (a ) c Φ(z ) ≈ ϕ (a )Φ(z)
0 0 i i i 0 i A 0 i i A 0
(cid:16) Xi (cid:17) Xi Xi
so Φ(a z) = ϕ (a )Φ(z) and similarly Φ(zb ) = Φ(z)ϕ (b ) for b ∈
0 A 0 0 B 0 0
B0. For arbitrary a ∈ A we may approximate a ≈ c a with a ∈ A0
i i i i
and we see that
P
Φ(az) ≈ c Φ(a z) = ϕ c a Φ(z) ≈ ϕ (a)Φ(z)
i i A i i A
X (cid:16)Xi (cid:17)
So Φ(az) = ϕ (a)Φ(z) for all a ∈ A and similarly Φ(zb) = Φ(z)ϕ (b)
A B
for all b ∈ B. We already know that Φ is injective since it preserves the
norm, so we only have to show that it is surjective. To see this note
that
Φ(X) = span Φ (X0) = span(Y0) = Y
0
So we have established that Φ is a linear isomorphism from X to Y
(cid:0) (cid:1)
which preserves the left and right actions and the inner product, in
(cid:3)
other words Φ is a correspondence isomorphism.
GivenaHilbertmodule(X,A),wedefinethelinking algebraof(X,A)
tobetheC∗-algebraL(X) := K(X⊕A). Therearecomplimentary pro-
∼ ∼
jections p and q in M(L(X)) such that pL(X)p = K(X), pL(X)q = X,
∼
and qL(X)q = A. This gives L(X) the following block matrix decom-
postion:
K(X) X
L(X) =
X A
(cid:20) (cid:21)
6 ADAMMORGAN
1 0 0 0
with p = and q = . The benefit of using linking algebras
0 0 0 1
(cid:20) (cid:21) (cid:20) (cid:21)
is that thealgebraic properties of X areencoded into the multiplicative
structure of L(X):
0 x 0 0 0 xa
(1) · =
0 0 0 a 0 0
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
∗
0 x 0 y 0 0
(2) · =
0 0 0 0 0 hx,yi
A
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
∗
0 x 0 y Θ 0
(3) · = x,y
0 0 0 0 0 0
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
Givenacorrespondence X overA, aToeplitz representation of(X,A)
in a C∗-algebra B is a pair (ψ,π) where ψ : X → B is a linear map
and π : A → B is a homomorphism satisfying:
ψ(xa) = ψ(x)π(a)
∗
π hx,yi = ψ(x) ψ(y)
A
ψ(ax) = π(a)ψ(x)
(cid:0) (cid:1)
By C∗(ψ,π) we shall mean the C∗-subalgebra of B generated by the
images of ψ and π in B. There is a unique (up to isomorphism) C∗-
algebra T , called the Toeplitz algebra of X, which is generated by a
X
universal Toeplitz representation (i ,i ).
X A
Let X⊗n denote the n-fold internal tensor product (see [Lan95] for
information on internal tensor products) of X with itself. By conven-
tion, we let X⊗0 = A. Let (ψ,π) be a Toeplitz representation of X
in B. Define the map ψn : X⊗n → B for each n ∈ N as follows: let
ψ0 = π, ψ1 = ψ, and set ψn(x⊗y) = ψ(x)ψn−1(y) (where x ∈ X and
y ∈ X⊗n−1) for each n > 1 .
We summarize Proposition 2.7 of [Kat04] as follows: Let (ψ,π) be a
Toeplitz representation of a correspondence X over A. Then
C∗(ψ,π) = span{ψn(x)ψm(y)∗ : x ∈ X⊗n,y ∈ X⊗m}
By Lemma 2.4 of [Kat04], for each n ∈ N there is a homomorphism
ψ(n) : K(X⊗n) → B such that:
(1) π(a)ψ(n)(k) = ψ(n)(φ(a)k) for all a ∈ A and all k ∈ K(X⊗n).
(2) ψ(n)(k)ψ(x) = ψ(kx) for all x ∈ X and all k ∈ K(X⊗n)
We define the Katsura ideal of A to be the ideal:
J = {a ∈ A : φ(a) ∈ K(X) and ab = 0 for all b ∈ ker(φ)}
X
CUNTZ-PIMSNER ALGEBRAS AND TWISTED TENSOR PRODUCTS 7
Whereφistheleftaction. ThisisoftenwrittenasasJ = φ−1 K(X) ∩
X
⊥
ker(φ) . In many cases of interest, the left action of a correspon-
(cid:0) (cid:1)
dence is injective and implemented by compact operators. In this case
(cid:0) (cid:1)
we have J = A.
X
A Toeplitz representation is said to be Cuntz-Pimsner covariant if
ψ(1) φ(a) = π(a) for all a ∈ J . The Cuntz-Pimsner algebra O is
X X
the quotient of T by the ideal generated by
X
(cid:0) (cid:1)
(1)
{i φ(a) −i (a) : a ∈ J }
X A X
It can be shown that OX(cid:0) is g(cid:1)enerated by a universal Cuntz-Pimsner
covariant representation (k ,k ).
X A
2.2. Actions and Coactions of Discrete Groups. Working with
actions andcoactions of locallycompact groupsand their crossed prod-
ucts can be somewhat complicated, for a general reference we suggest
the appendix of [EKQR02]. Restricting attention to discrete groups
affords many simplifications. In this section, we will summarize some
of these simplifications.
Proposition 2.3. Let G be a discrete group and let α be an action
of G on a C∗-algebra A. Let (i ,i ) be the canonical representation
A G
of the system (A,G,α) in the reduced crossed product A⋊ G. Then
α,r
A ⋊ G is the closed linear span of elements of the form i (a)i (s)
α,r A G
where a ∈ A and s ∈ G. These have the following algebraic properties:
(4) i (a)i (s) i (b)i (t) = i aα (b) i (st)
A G A G A s G
(5) (cid:0) (cid:1)i(cid:0)A(a)iG(s) (cid:1)∗ = iA(cid:0)αs−1(a(cid:1)∗) iG(s−1)
Proof. The fact that th(cid:0)e i (a)i (s(cid:1)) dense(cid:0)ly span A(cid:1)⋊ G follows from
A G α,r
the definition of the crossed product. The algebraic properties follow
easily from the fact that i (a)i (s) = i (s)i α (a) which we obtain
A G G A s
from i (s)∗i (a)i (s) = i α (a) . (cid:3)
G A G A s (cid:0) (cid:1)
Proposition 2.4. Let (γ,α(cid:0)) be a(cid:1)n action of a discrete group G on a
correspondence (X,A). Let (i ,i ,iX,iA) be the canonical representa-
X A G G
tion of the system in the crossed product (X ⋊ G,A⋊ G). Then
γ,r α,r
X⋊ G is the closed linear span of i (x)iX(s) where x ∈ X and s ∈ G.
γ,r X G
These satisfy the following algebraic properties:
(6) i (x)iX(s) i (a)iA(t) = i xα (a) iX(st)
X G A G X s G
(7) iX(x(cid:0))iXG(s),iX(y(cid:1))(cid:0)iYG(t) A⋊α,rG(cid:1) = iA(cid:0)αs−1 hx(cid:1),yiA iAG(s−1t)
(8) (cid:10) i (a)iA(s) i (x(cid:11))iX(t) = i (cid:0)aγ ((cid:0)x) iX(st(cid:1))(cid:1)
A G X G X s G
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
8 ADAMMORGAN
Proof. The fact that the i (x)iX(s) densely span X⋊ G follows from
X G γ,r
the definition of the crossed product. To understand the algebraic
properties, recall from Lemma 3.3 of [EKQR02] that L(X ⋊ G) ∼=
γ,r
L(X) ⋊ G where ν is the coaction on L(X) induced by (γ,α). (6)
ν,r
and (7) are then easily deduced by applying the previous proposition
to L(X)⋊ G. (8) follows from the fact that the left action must be
ν,r
(cid:3)
covariant with respect the action.
Corollary 2.5. The sets
(X ⋊ G) := {i (x)iX(s) : x ∈ X,s ∈ G}
γ,r 0 X G
(A⋊ G) := {i (a)iA(s) : a ∈ A,s ∈ G}
α,r 0 A G
form a generating system for X ⋊ G in the sense of Definition 2.1.
γ,r
Thesimplification ofcrossed productsbycoactionsofdiscrete groups
come from the realization that coactions by discrete groups can be
viewed as gradings. This idea is presented in detail in [Qui96], but we
briefly summarize the main points in the next two propositions. We
refer the reader to [Qui96] for the proofs.
Proposition 2.6. Let δ : A → M A ⊗ C∗(G) be a coaction of a
discrete group G on a C∗-algebra A. Then A = span{A } where
s s∈G
(cid:0) (cid:1)
A = {a ∈ A : δ(a) = a⊗u }. Furthermore,
s s
(9) A ·A ⊆ A
s t st
∗
(10) As = As−1
Proposition 2.7. Let δ be a coaction of a discrete group G on a C∗-
algebra A and let (j ,j ) be the canonical representations of A and
A G
c (G) in the crossed product A⋊ G. Then A⋊ G is densely spanned
0 δ δ
by elements of the form j (a )j (f) where a ∈ A , f ∈ c (G) and
A s G s s 0
These satisfy the following relations:
(11) jA(a)jG(f) jA(as)jG(g) = jA(aas)jG λs−1(f)g
∗ ∗
(12) (cid:0) (cid:1)j(cid:0)A(as)jG(f) (cid:1) = jA(as)jG λ(cid:0)s(f) (cid:1)
where λ denotes left t(cid:0)ranslation by(cid:1)s on c (G). (cid:0) (cid:1)
s 0
Applying these propositions to linking algebras helps us to under-
stand coactions on correspondences:
Proposition 2.8. Let (σ,δ) be a coaction of a discrete group G on a
Hilbert module (X,A). Then X = span{X } where X = {x ∈ X :
s s∈G s
CUNTZ-PIMSNER ALGEBRAS AND TWISTED TENSOR PRODUCTS 9
σ(x) = x⊗u }. Further:
s
(13) X ·A ⊆ X
s t st
(14) hxs,xtiA ∈ As−1t
(15) A ·X ⊆ X
s t st
Proof. Let ε be the induced coaction on L(X). Then we have a grading
L(X) = span{L(X) } . Since p and q are in fixed points of the
s s∈G
coaction, if z ∈ L(X) then qzq ∈ L(X) and pzq ∈ L(X) . Recall
s s s
0 0
∼
that the restriction of ε to qL(X)q = = A is δ. Thus if a is the
0 A
(cid:20) (cid:21)
element of A corresponding to qzq then ε(qzq) = (qzq)⊗u if and only
s
0 0
∼
if δ(a) = a ⊗ u . Thus qL(X) q = = A . Similar reasoning
s s 0 A s
s
(cid:20) (cid:21)
0 X
shows that pL(X) q = s ∼= X . (13) and (14) then follow from
s 0 0 s
(cid:20) (cid:21)
multiplication in L(X) together with the grading of L(X). (15) follows
from the fact that the left module action is covariant with respect to
(cid:3)
the coaction.
Lemma 3.4 of [EKQR02] shows that L(X ⋊ G) ∼= L(X)⋊ G. This
σ ε
fact, together with the preceding propositions, gives us the following
characterization of X ⋊ G in the case where G is discrete:
σ
Proposition 2.9. Let (σ,δ) be a coaction of a discrete group G on a
correspondence (X,A). Let (j ,j ,jX,jA) be the canonical represen-
X A G G
tation of the system in the correspondence (X ⋊ G,A ⋊ G). Then
σ δ
X⋊ G is densely spanned by elements of the form j (x )jX(f) where
σ X s G
x ∈ X and f ∈ c (G). These elements satisfy the following relations:
s s 0
(16) jX(x)jGX(f) jA(as)jGA(g) = jX(xas)jGX(λs−1(f)g)
(17) jX(x(cid:0)s)jGX(f),jX((cid:1)x(cid:0)t)jGX(g) A⋊ G(cid:1) = jA hxs,xtiA jGA λt−1s(f)g
δ
(18) (cid:10) jA(a)jGA(f) jX(xs)jG(cid:11)X(g) = jX(cid:0)(axs)jGX λ(cid:1)s−1(cid:0)(f)g (cid:1)
Corollary(cid:0)2.10. The s(cid:1)e(cid:0)ts (cid:1) (cid:0) (cid:1)
(X ⋊ G) := {j (x )jX(f) : x ∈ X ,f ∈ c (G)}
σ 0 X s G s s 0
(A⋊ G) := {j (a )jA(f) : a ∈ A ,g ∈ c (G)}
δ 0 A s G s s 0
form a generating system for X ⋊ G.
σ
2.3. Quantum Groups. There are many different notions of a quan-
tum group. Our quantum groups will be quantum groups generated
by modular multiplicative unitaries. We will briefly recall some of the
10 ADAMMORGAN
basic facts about these quantum groups and refer the reader to [Tim08]
for a more in depth overview of the subject.
Definition 2.11. Given a separable Hilbert space H, a multiplicative
unitary W is a unitary operator on H⊗H such that
(19) W W = W W W ∈ U(H⊗H⊗H)
23 12 12 13 23
where W indicates that we are applying W to the ith and jth factors
ij
of H and leaving the other fixed. Equation 19 is sometimes referred to
as the pentagon equation.
W is called modular if there exist (possibly unbounded) operators Q
and Q on H and a unitary W ∈ U(H⊗H) (where H is the dual space
of H) such that b
f
(1) Q and Q are positive and self-adjoint with trivial kernels
(2) W∗(Q⊗Q)W = Q⊗Q
(3) hbη′ ⊗ ξ′,W(η ⊗ ξ)i = hη ⊗ Qξ′,W(η′ ⊗ Q−1ξ)i for all ξ ∈
Domb(Q−1), ξ′ ∈ Dbom(Q), and η,η′ ∈ H
f
Example 2.12 (Example 7.1.4 of [Tim08]). Let G be a locally compact
group. We can identify L2(G) ⊗ L2(G) with L2(G × G) and define
W ∈ B L2(G)⊗L2(G) by
G
(cid:0) (W(cid:1) Gζ)(s,t) := ζ(s,s−1t)
Then W is a multiplicative unitary.
G
Theorem 2.13 (Theorem 2.7 of [MRW14]). For a modular multiplica-
tive unitary W ∈ U(H⊗H), set
S := span{(ω ⊗id )W : ω ∈ B(H) }
H ∗
S := span{(id ⊗ω)W : ω ∈ B(H) }
H ∗
Then:
b
• S and S are separable, nondegenerate C∗-subalgebras of B(H).
• W ∈ U(S ⊗ S) ⊆ U(H ⊗ H). When we wish to view W as a
unitarybmultiplier of S ⊗S we will denote it by WS and refer
to it as tbhe reduced bicharacter of S.
• There are unique hombomorphisms ∆ ,∆ : S → S ⊗ S such
S S
that
(idSb⊗∆S)WS = W1S2W1S3 ∈ U(S ⊗b S ⊗S)
(∆ ⊗id )WS = WSWS ∈ U(S ⊗S ⊗S)
S S 23 13 b
• (S,∆ ) and (S,∆ ) are C∗-bialgebras
S b S b b
b b