ingly, we have the following result which guarantees that the abstract and concrete notions of Banach algebra coincide.
Cayley’s theorem for Banach algebras: Every Banach algebra acts faithfully on a Banach space by bounded linear operators such that the norm on agrees with the operator norm.
Proof. Consider the action of on itself by left multiplication; we denote the operator by . This is clearly an injective algebra homomorphism (Cayley’s theorem for algebras). Since we clearly have ; moreover, since we in fact have .
Recall that the spectrum consists of all scalars such that is not invertible. This definition turns out to be very fruitful for Banach algebras in particular due to the following results.
Proposition: Let be an element of a Banach algebra . Then the limit exists.
Proof. Let . Then for any we can find such that , hence . It follows that for we have
hence that . Since this is true for every , we conclude that .
Proposition: Let be such that . Then is invertible.
Proof. The geometric series converges absolutely and is a two-sided inverse for .
Corollary: Let . Then .
Proof. If , then is invertible by the above.
Accordingly, we define the spectral radius
This is the radius of the smallest circle about the origin containing the spectrum of . The above shows that we always have .
For a Banach algebra let , the general linear group of , denote its group of units.
Proposition: is open. The inversion map is continuous (making a topological group).
Proof. By translation it suffices to show that an open neighborhood of the identity in lies in . But this is clear since the open ball of radius, say, about the identity lies in by the above.
Let . Note the identity , which gives
It follows that for fixed we can make arbitrarily small by making sufficiently small.
Corollary: The spectrum is compact.
Proof. The resolvent is defined on the complement of the spectrum (the resolvent set ) and has image in . By the above, is continuous, hence maps open sets to open sets. Since is open in , it follows that is open in , so the spectrum is closed. Since the spectrum is also bounded, it is compact by Heine-Borel.
Proposition: Let be a Banach algebra and a (left, right, two-sided) ideal of . Then:
- The closure of is also a (left, right, two-sided) ideal.
- If is proper, then so is .
- If is two-sided, closed, and proper, the quotient is also a Banach algebra and the map is a morphism of Banach algebras.
- If is maximal, it is already closed.
Proof. : is the closure of a subspace, hence also a subspace, hence closed under addition and scalar multiplication. If is a Cauchy sequence converging to , then , hence is a Cauchy sequence converging to , so is closed under left multiplication by elements of if is and similarly for right multiplication.
: An ideal is proper if and only if it contains no invertible elements if and only if it does not contain the identity (and this is true for left, right, and two-sided ideals). Since the open ball of radius about the identity consists of invertible elements, it must be disjoint from any proper ideal , so cannot contain the identity and hence must also be proper.
: If is two-sided, closed, and proper, the quotient is a Banach space and an algebra, and the map is a weak contraction and an algebra homomorphism. It remains to show that the quotient norm on satisfies , but this follows from the fact that the cosets contain the possible products of elements in the cosets , hence the infimum of the norms in the former is at most the infimum of the norms in the latter.
This already shows that the quotient norm on satisfies , since implies and the quotient map is a weak contraction. A direct argument is to observe that the open ball of radius about consists only of invertible elements, hence is disjoint from .
: Follows directly from .
Corollary: Any algebra homomorphism is a morphism of Banach algebras.
This is one of several results in Banach algebra theory to the effect that an algebra homomorphism between certain Banach algebras is automatically continuous; see, for example, Section I.5 of Dales