# C*-Algebras

**Definition 1**. A **Banach algebra** is a complex algebra which is a Banach space and satisfies subnormality:

.

A ***-algebra** is a complex algebra with a unary operation * that satisfies , , , and . A **Banach *-algebra** is a Banach algebra and a *-algebra that satisfies . A **C*-algebra** is a Banach *-algebra that satisfies . A map between two *-algebras is a ***-homomorphism** if .

**Proposition 2.** Any nonunital Banach algebra has a unitization .

*Proof.* Let . Let , and send . Define , , and . Clearly is a Banach space. We also have

The unit is : .

In fact, if is a nonunital Banach *-algebra, then there is a unitization of which is also a Banach *-algebra. We simply use and define . And if is a C*-algebra, then we modify the norm on to .

**Definition 3.** Let be a Banach algebra and . The **spectrum of** is the set

.

Clearly for nonunital , for all .

**Lemma 4**. Let be a unital Banach algebra. If , then exists and is defined by

.

*Proof.* Denote by . Then

Since , the series is absolutely convergent and hence convergent. So the two sums cancel–leaving .

**Corollary 5.** If , then is a unit.

**Proposition 6.** Let be a unital Banach algebra and .

- is a nonempty compact subset of .
- .

*Proof. *(1) By Heine-Borel, it suffices to show is closed and bounded for all . Let denote the set of units of and a scalar be called a **regular point of ** if . Let be a regular point. Then

.

Since is open, there is a neighborhood . Moreover, since the function defined by is continuous, is open in . Thus the set of regular points is open–implying that the spectrum is closed.

Now if , then is a regular point by the previous claim. Hence .

Nonemptiness and (2) are omitted.

**Corollary 7 (Gelfand-Mazur)**. If is a Banach division algebra, then .

**Proposition 8.** Let be a C*-algebra and be self-adjoint (). Then .

*Proof.* From the C*-axiom we have . Iterating this yields . Hence by (2) of Proposition 6 and the C*-axiom,

.

Since is self-adjoint for all , this says that the norm of a C*-algebra is uniquely determined: .

**Theorem 9 (Gelfand-Naimark)**. Let be a commutative unital C*-algebra and be a closed subset of the unit ball of the dual space . Then the map defined by , called the **Gelfand transform**, is an isometric *-isomorphism.

[1] Bachman, George and Lawrence Narici. *Functional Analysis*. Dover Publications. 2000.

[2] Blackadar, Bruce. *Operator Algebras*. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.