All the spaces, algebras in this notes are assumed to be over C.
1. C∗-algebra
A Banach ∗-algebra is a complex Banach algebra A together with a conjugate linear involution ∗ (called the adjoint) which is an anti-linear isomorphic, That is, for all a, b ∈ A, and λ ∈ C, such that
(1) (a + b)∗ = a∗+ b∗ (2) (λa)∗ = λa∗ (3) a∗∗= (a∗)∗ = a (4) (ab)∗ = b∗a∗.
A C∗-algebra is a Banach ∗-algebra with
ka∗ak = kak2.
This implies that kak ≤ ka∗k. Similarly, we find ka∗k ≤ kak. Hence kak = ka∗k. This also implies that kaa∗k = kak2.
Example 1.1. Let X be a compact metric space and C(X) be the space of all complex- valued continuous functions on X. Define
kf k∞= sup
x∈X
|f (x)|.
Then C(X) is a Banach algebra. Moreover if we set f∗ = f , then C(X) becomes a commu- tative C∗-algebra.
Example 1.2. Let H be a Hilbert space. The space of bounded operators on H denoted by B(H) is a C∗-algebra. The normed on B(H) is given by
kAkB(H) = sup
kxk=1
kAxkH.
Definition 1.1. A normed closed self-adjoint sub algebra of B(H) is called a concrete C∗-algebra.
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