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.
 Bachman, George and Lawrence Narici. Functional Analysis. Dover Publications. 2000.
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.