Download INTRODUCTION TO C* ALGEBRAS - I Introduction : In this talk, we

INTRODUCTION TO C* ALGEBRAS - I
DR. PRAHLAD VAIDYANATHAN
Introduction : In this talk, we introduce the notion of a C* algebra (a.k.a. Operator
Algebra). The theory of C* algebras has its roots in functional analysis (where the basic
object is a normed vector space). C* algebras provide an interesting place where analytic
notions (such as limits and differentiability) have been married to algebra (group actions,
ring theory) and topology (homological algebra, homotopy theory), thereby helping to
answer a variety of problems.
The theory of C* algebras was developed in the late 1920s by Gelfand and Naimark (with
a view to solving problems in Harmonic Analysis) and by John Von Neumann (with a
view to establishing the mathematical foundations of quantum physics).
Broadly speaking, a C* algebra is a ring which carries a metric topology, such that the
ring operations are continuous (in the same way that the group operations are continuous
in a topological group). We begin with two examples, and proceed to give the definition.
1. The Space of Continuous Functions
Let X be a compact, Hausdorff space. Consider the space of complex-valued continuous
functions on X
C(X) = {f : X → C : f is continuous}
Note that C(X) is a complex vector space under the point wise operations
(αf )(x) = αf (x)
and
(f + g)(x) = f (x) + g(x)
Furthermore, C(X) carries the structure of a normed linear space with the sup norm
kf k = sup |f (x)|
x∈X
Also, C(X) is an algebra under point-wise multiplication.
It turns out, the algebraic operations on C(X) are continuous with respect to the norm
because of the following inequalities
kf + gk ≤ kf k + kgk
and
kf gk ≤ kf kkgk
Definition 1.1. A (complex) algebra A equipped with a norm such that
kabk ≤ kakkbk ∀ a, b ∈ A
is called a normed algebra. Furthermore, if A is complete with respect to the norm (ie.
cauchy sequences converge), then A is called a Banach Algebra.
(Note: The condition kabk ≤ kakkbk is equivalent to the fact that the multiplication
operation is bi-continuous)
Date: Sept 13, 2012.
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Note that C(X) is a Banach algebra. We will give more examples in the next section. In
order to motivate the definition of a C* algebra, we examine the correspondence between
the topological structure of X and the Banach algebraic structure of C(X) :
f
(1) If X →
− Y is a continuous map, then
ϕf : C(Y ) → C(X)
g 7→ g ◦ f
is a homomorphism of rings. Hence the map
X 7→ C(X)
is a contravariant functor.
(2) Furthermore, if ϕ : C(Y ) → C(X) is a unital ring homomorphism, then there is
a continuous function f : X → Y such that
ϕ = ϕf
(3) If x ∈ X, then the map
ψx : C(X) → C
f 7→ f (x)
is linear and respects the multiplication operator (ie. ψx is a multiplicative linear
functional). Furthermore, every multiplicative linear functional is of the form ψx
for some x ∈ X. Hence, the set
Ω := {ψ : C(X) → C : ψ is multiplicative and linear}
can be identified with X. Furthermore, this set carries a natural topology, the
so-called weak-* topology, where
ψn → ψ
⇔
ψn (f ) → ψ(f ) ∀ f ∈ C(X)
It turns out that the map
x 7→ ψx
X→Ω
gives a homeomorphism between the topology of X and the weak-* topology on Ω.
Hence, we may completely recover the topological structure of X from the
Banach algebraic structure of C(X).
2. Bounded operators on a Hilbert Space
Suppose H is a complex Hilbert space (think H = Cn or H = l2 ). A bounded operator
on H is a continuous linear map T : H → H. We denote the space of all such operators
by B(H). Note that B(H) is itself a vector space, and has a multiplication given by
composition of operators
T ·S =T ◦S
Also, B has a natural operator norm defined on it
kT k = sup{kT xk : x ∈ H}
It is easy to check that
kT Sk ≤ kT kkSk
and hence B(H) is a normed algebra (in fact, it is a Banach algebra, since it is complete
with respect to the operator norm).
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Suppose T ∈ B(H), we define the adjoint T ∗ by the formula
< T (x), y >=< x, T ∗ y >
In a course in functional analysis, one learns that T ∗ behaves like a complex-conjugate
of T in that there are many interesting properties of T that one can obtain from T ∗ . For
instance,
(1) (T ∗ )∗ = T
(2) λ is an eigen-value of T if and only if λ is an eigen-value for T ∗
(3) kT k = kT ∗ k = kT ∗ T k1/2
Furthermore, if T is normal (T ∗ T = T T ∗ ), self-adjoint (T = T ∗ ), or unitary (T ∗ =
T −1 ), then we can say a lot more about the operator (for instance, we may be able to
understand its eigen-values/spectrum much better in this case). Suffice to say, there is
a deep connection between T and T ∗ , and in many problems arising in single-operator
theory, it helps a great deal to consider both T and T ∗ simultaneously.
3. Definition
A C* algebra is a specific type of Banach algebra. It is a way of combining the above
two examples into a unified theory that seeks to address problems that arise in operator
theory using topology, and vice-versa.
Definition 3.1. A C* algebra is a Banach algebra A together with an involution operation a 7→ a∗ such that
(a∗ )∗ = a
(a + b)∗ = a∗ + b∗
(αa)∗ = αa∗
(ab)∗ = b∗ a∗
ka∗ ak = kak2
(C* identity)
We discuss this definition, and give some examples in the following lecture.
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