Some basic concepts
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Group C ∗ algebras
Let G be a locally compact group. We have the left invariant Haar measure
µ on it, and let ∆ be the modular function defined by
dµ(g −1 ) = ∆(g)−1 dµ(g),
i.e. ∆(·) is the (almost everywhere strictly positive) Radon-Nikodym derivative dµ(g)dµ(g −1 ) between the equivalent left and right invariant Haar measures.
Consider the Banach space L1 (G). We want to make it into a Banach
algebra with involution. Define the binary operation ∗ :
Z
(φ ∗ ψ)(g) :=
φ(h)ψ(h−1 g)dµ(h), φ, ψ ∈ L1 (G).
G
Also, define an involution ] by f ] (g) := ∆(g)−1 f (g −1 ).
Exercise
Check that the above makes L1 (G) into a Banach algebra with involution.
(Hint: Use the invariance of Haar measure to verify associativity of ∗, and
also kφ ∗ ψk1 ≤ kφk1 kψk1 .)
Lemma 1.1 There is a net fi in L1 (G) such that kfi k1 ≤ 1 for all i and
for every f ∈ L1 (G), kf ∗ fi − f k1 and kfi ∗ f − f k1 go to zero.
By replacing fi by (fi + fi] )/2, one can arrange fi to be self-adjoint, and
also observe that kfi kkf k ≥ kfi f k → kf k for all f , so that kfi k is bounded
away from zero and hence we can also arrange kfi k = 1. Such a choice, i.e.
with kfi k = 1 and fi] = fi will be the standard convention from now on. We
shall refer to such a net fi as an approximate unit or approximate identity.
Exercise: For a discrete group G, L1 (G) ≡ l1 (G) has an identity χe ,
where χe denotes the characteristic function of the identity element of the
group.
Now, we want to show how to find ∗-homomorphism of L1 (G) into B(H)
for Hilbert space H.
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Theorem 1.2 Given a strongly continuous unitary representation g 7→ Ug
of the group G into some Hilbert space H, there is a ∗-homomorphism πU :
L1 (G) → B(H) given by:
Z
f (g)Ug dµ(g).
πU (f ) :=
G
This is a nondegenerate representation in the sense that vectors of the form
πU (f )ξ where f ∈ L1 (G) and ξ ∈ H form a total subset of H.
Conversely, any nondegenerate ∗-homomorphism of L1 (G) arises in this
way.
Thus, there are plenty of Hilbert space representations of the involutive
Banach algebra L1 (G), and we also know (by the analysis of spectral radius)
that any ∗-homomorphism from an involutive Banach algebra into B(H)
must be contractive. This makes the following well-defined and finite:
kf ku := sup kπU (f )k,
(π,H)
for f ∈ L1 (G), where supremum is taken over all Hilbert space representations of L1 (G). Clearly, kf ku ≤ kf k1 . In fact, it is strictless less unless G is
trivial, as L1 (G) is never a C ∗ algebra unless G is trivial. Exercise Check
that k · ku is a C ∗ -norm.
We define the free or full group C ∗ algebra C ∗ (G) to be the completion
of L1 (G) under the norm k · ku . On the other hand, there is a special representation πU for the left regular representation U of G given by Ug = Lg
in L2 (G). The completion of πU (L1 (G)) in the norm of B(L2 (G)) is defined
to be the reduced group C ∗ algebra Cr∗ (G).
In general, C ∗ (G) is bigger than Cr∗ (G), but these two coincide for a class
of groups called amenable groups for which there is a left-invariant state,
called a left-invariant mean, on L∞ (G)) . Such groups include all abelian
and compact groups. However, for the free group with two generators, these
two differ!
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Multiplier algebra
For a C ∗ algebra A (possibly non-unital), its multiplier algebra, denoted by
M(A), is defined as the maximal C ∗ algebra which contains A as an essential two-sided ideal, that is, A is an ideal in M(A) and for y ∈ M(A),
ya = 0 for all a ∈ A implies y = 0. In case A is unital, one has M(A) = A
2
and for A = C0 (X) where X is a noncompact, locally compact Hausdorff
space, M(A) = C(X̂), where X̂ denotes the Stone-Ĉech compactification
of X. The norm of M(A) is given by kxk := supa∈A,kak≤1 {kxak, kaxk}.
Furthermore, there is a canonical locally convex topology, called the strict
topology on M(A), which is given by the family of seminorms {k.ka , a ∈
A}, where kxka := Max(kxak, kaxk), for x ∈ M(A). We say that a C ∗ algerbra A ⊆ B(h) for some Hilbert space H is nondegenerate if for u ∈ H,
au = 0 for all a ∈ A implies that u = 0. It is easy to show that A ⊆ B(H)
is nondegenerate if and only if {au, a ∈ A, u ∈ H} is total in H. This
is also equivalent to the condition that any approximate identify of A converges weakly to the identity of B(H). Given a nondegenerate C ∗ -subalgebra
A ⊆ B(H), we have that M(A) ∼
= {x ∈ B(H) : xa, ax ∈ A, for all a ∈ A}.
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Hilbert module
A Hilbert space is a complex vector space equipped with a complex valued
inner product. A natural generalization of this is the concept of Hilbert
module, which has become quite an important tool of analysis and mathematical physics in recent times.
Definition 3.1 Given a ∗-subalgebra A ⊆ B(h) (where h is a Hilbert space),
a semi-Hilbert A-module E is a right A-module equipped with a sesquilinear map h., .i : E × E → A satisfying hx, yi∗ = hy, xi, hx, yai = hx, yia and
hx, xi ≥ 0 for x, y ∈ E and a ∈ A. A semi-Hilbert module E is called a
pre-Hilbert module if hx, xi = 0 if and only if x = 0; and it is called a
Hilbert module if furthermore A is a C ∗ -algebra and E is complete in the
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norm x 7→ khx, xik 2 where k.k the C ∗ norm of A.
It is clear that any semi-Hilbert A-module can be made into a Hilbert module
in a canonical way : first quotienting it by the ideal {x : hx, xi = 0} and
then completing the quotient.
Definition 3.2 Let E and F be two Hilbert A-modules. We say that an
C-linear map L from E to F is adjointable if there exists a C-linear map
L∗ from F to E such that hL(x), yi = hx, L∗ (y)i for all x ∈ E, y ∈ F . We
call L∗ the adjoint of L. The set of all adjointable maps from E to F is
denoted by L(E, F ). In case E = F , we write L(E) for L(E, E).
The first important fact to be noted is that an adjointable map is automatically A-linear and norm-bounded.
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Proposition 3.3 Let E, F be Hilbert A-modules and L ∈ L(E, F ). Then
we have the following :
(i) Both L and its adjoint L∗ are A-linear.
(ii) L is a norm-bounded map from E to F , viewed as Banach spaces.Similarly,
L∗ is bounded.
(iii) L∗ is the unique C-linear map satisfying hL(x), yi = hx, L∗ (y)i for all x ∈
E, y ∈ F. Furthermore, (L∗ )∗ = L.
(iv) If E1 , E2 , E3 are Hilbert A-modules and L1 ∈ L(E1 , E2 ), L2 ∈ L(E2 , E3 ),
then L2 L1 := L2 ◦ L1 ∈ L(E1 , E3 ) with (L2 L1 )∗ = L∗1 L∗2 .
We define a norm on L(E, F ) by kLk := supx∈E,kxk≤1 kL(x)k. We have
already proven that for L ∈ L(E, F ), kLk is indeed finite. Clearly, L(E) is
a C ∗ -algebra with this norm.
The topology on L(E, F ) given by the family of seminorms {k.kx , k.ky :
1
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x ∈ E, y ∈ F } where ktkx = khtx, txi 2 k and ktky = kht∗ y, t∗ yi 2 k, is known
as the strict topology. For x ∈ E, y ∈ F , we denote by |y >< x| ≡ θx,y
the element of L(E, F ) defined by θx,y (z) = yhx, zi (z ∈ E). The L(E)norm-closed subset generated by A-linear span of {θx,y : x ∈ E, y ∈ F } is
called the set of compact operators and denoted by K(E, F ). It should
be noted that these objects need not be compact in the sense of compact
operators between two Banach spaces, though this abuse of terminology has
become standard. It is known that K(E, F ) is dense in L(E, F ) in the strict
topology. In case F = E, we denote K(E, F ) by K(E). It is clear that K(E)
is a C ∗ algebra which is dense in the C ∗ -algebra L(E) with respect to the
strict topology. Moreover, M(K(E)) ∼
= L(E).
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Tensor and free product
Let (Ai )i∈I be a family of unital C ∗ algebras Consider the category with
objects (B, {φi }i∈I where B is a unital C ∗ algebra and φi : Ai → B unital
∗-homomorphisms. For two such objets X = (B, {φi }) and Y = (C, {φ0i }),
Mor(X, Y ) consists of unital ∗-homomorphisms π : B → C satisfying π ◦φi =
φ0i for all i.
Theorem 4.1 There exists a (unique upto isomorphism) universal object
in the above category.
This universal object is denoted by (A = ∗i Ai , ψi }, called the unital C ∗
algebra free product of the given family. ψi : Ai → A gives canonical
embedding of Ai into A, and A is generated as a C ∗ algebra by the images
of these embeddings.
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Remark 4.2 It is a direct consequence of the above definition that given a
family of unital ∗-homomorphisms φi from Ai to B, there exists a unique
unital ∗-homomorphism φ = ∗i φi from ∗i Ai to B such that φ ◦ ψi = φi for
all i.
Remark 4.3 Also, for discrete groups {Gi }i∈I , one has C ∗ (∗i Gi ) ∼
= ∗i C ∗ (Gi ).
Next we come to tensor product. We assume familiarity with algebraic
tensor product as well as tensor product of Hilbert spaces. For two algebras
A, B, we will denote the algebraic tensor product of them by the symbol
A⊗alg B. When they are C ∗ algebras, there is usually more than one possible
choice of a norm on A ⊗alg B satisfying ka ⊗ bk = kakkbk, and also that the
completion with respect to the norm is a C ∗ algebra. We will work with
the so called injective tensor product, (also called spatial or minimal
tensor product ) that is, the completion of A⊗alg B with respect to the norm
given by
k
n
X
ai ⊗ bi k := sup k
i=1
X
π1 (ai ) ⊗ π2 (bi )kB(H1 ⊗H2 ) ,
i
where ai is in A, bi is in B and the supremum runs over all possible choices
of (π1 , H1 ), (π2 , H2 ), where H1 , H2 are Hilbert spaces and π1 : A → B(H1 )
and π2 : A2 → B(H2 ) are -homomorphisms. The norm k · kB(H1 ⊗H2 ) is the
operator norm on B(H1 ⊗ H2 ).
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Hopf algebra
Cosndier a finite group G and the algebra C(G) of complex valued functions
on it. This is obviously a ∗-algebra by the pointwise operations, but this algebra structure does not use the group structure at all. To encode the group
structure, we can define ∗-homomorphism ∆ : C(G) → C(G) ⊗ C(G) =
C(G × G) by ∆(f )(g, h) = f (gh), where gh is the group multiplication of
two elements g and h. The associativity of the group multiplication translates into the following property (called co-associativity) of ∆:
(∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆.
(1)
Moreover, the identity element of the group, say e, can be encoded into a
∗-homomorphic map : C(G) → C given by (f ) = f (e), and the inverse
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is encoded by κ : C(G) → C(G) given by κ(f )(g) = f (g −1 ). It is easy to
verify the following :
( ⊗ id) ◦ ∆ = (id ⊗ ) ◦ ∆ = id,
(2)
m ◦ (κ ⊗ id) ◦ ∆(f ) = m ◦ (id ⊗ κ) ◦ ∆(f ) = (f )1,
(3)
where m denotes the algebra multiplication from A⊗A to A. This motivates
the following definition:
Definition 5.1 A Hopf algebra over a field F is a tuple (Q, m, 1, ∆, , κ)
where B is a unital algebra over F , 1 being the identity element of B and m
the algebra multiplication map, ∆ : B → B ⊗B is an algebra homomorphism,
κ : B → F multiplicative linear map and κ : B → B is a linear map satisfying
(1),(2),(3) above. In case F is R or C, B is a ∗-algebra and the maps ∆, are ∗-homomorphism, we say that the Hopf algebra is a Hopf ∗ algebra.
It can be proved that κ is an anti-homomorphism, i.e. κ(ab) = κ(b)κ(a),
and moreover, for Hopf ∗ algebra, κ(κ(a∗ )∗ ) = a. We shall often write
∆(a) = a(1) ⊗ a(2) to abbreviate the expression of ∆ as a finite sum of
elements of the form b ⊗ c, b, c ∈ B.
Exercise: Dual of a Hopf algebra is again a Hopf algebra. For example, the
dual of C(G) is the group algebra CG.
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