This note is a collection of results on Banach algebras whose proofs do not require the machinery of integrating functions that take values in Banach spaces, and that do not require the algebra to be commutative.
2 Banach algebras and ideals
Unless we say otherwise, every vector space we talk about is taken to be over . A Banach algebra is a Banach space that is also an algebra satisfying for . We say that is unital if there is a nonzero element such that and for all , called a identity element. If is a Banach space, let denote the set of bounded linear operators , and let denote the set of compact linear operators (to be compact means that the image of each bounded set is precompact). is a unital Banach algebra. is a Banach algebra, but it is unital if and only if is finite dimensional.
An ideal of a Banach algebra is a vector subspace such that
We define the algebra quotient by
this makes sense because is an ideal. We define a seminorm on by
we call this the quotient seminorm. In the following theorem we show that if the ideal is closed then this seminorm is a norm.
If is a closed ideal in a Banach algebra , then is a Banach algebra with the quotient norm.
Suppose that . This means
Let with . That is, . Since is closed we get , and hence , showing that the seminorm on the quotient is in fact a norm.
Let be a Cauchy sequence and let be a subsequence with for all . We shall use the fact that is complete to prove that the sequence converges and therefore that is complete. We define a sequence in inductively as follows. Let be any element of . Suppose that , and let be any element of . We have
As this is an infimum and the inequality is strict, there is some for which , and we define . Thus defined, the sequence satisfies for all and hence is a Cauchy sequence (as the consecutive differences are summable). Thus converges to some , as is a Banach space. We have
which tends to as . Therefore . As is a Cauchy sequence a subsequence of which converges to , it follows that . We have shown that each Cauchy sequence in converges, and hence is complete. ∎
If is a Banach space, then is a closed ideal of the Banach algebra . Hence with the quotient norm, the quotient algebra is a Banach algebra. Let denote the set of bounded finite rank linear operators (to be finite rank means to have a finite dimensional image). is an ideal of , but it is closed if and only if is finite dimensional. The closure of is contained in . (If the closure is equal to we say that the Banach space has the approximation property.)
3 Left regular representation
A Banach algebra is in particular a Banach space, and thus is itself a Banach algebra with the operator norm. The left regular representation is the map defined by
It is apparent that is an algebra homomorphism, and for ,
showing that . If has identity element , then , and as we get . We’d like the norm on to satisfy , and we define
Check that is injective, from which it follows that is nondegenerate and is thus a norm on . On the other hand,
so and are equivalent norms on , and thus for most purposes anything we say using one can be said just as well using the other. We decree that if we are talking about a unital Banach algebra then its norm is such that . Then, is an isometry.
4 Invertible elements
If is a unital Banach algebra, we say that is invertible if there is some such that and , and we call the inverse of . We denote by the set of invertible elements of , and we call the multiplicative group of the Banach algebra. We now prove that a perturbation of norm of the identity element in a unital Banach algebra remains in the multiplicative group.11 1 Walter Rudin, Functional Analysis, second ed., p. 249, Theorem 10.7.
If is a unital Banach algebra, , and , then , and
Define . For we have
Thus is a Cauchy sequence and so has a limit . Because , we have
so . But
so . Therefore . One similarly shows that , and hence that , with . Furthermore,
In the above theorem we found that if then . This is an analog of the geometric series, and is called a Neumann series.
If is a Banach algebra and is a linear map satisfying
then is called a complex homomorphism on . That is, a complex homomorphism on is an algebra homomorphism . It is straightforward to prove that if is a unital Banach algebra and is a complex homomorphism on , then and for all . A linear functional on a Banach algebra need not be continuous, but in the following we prove that a nonzero complex homomorphism has operator norm 1, and in particular is continuous.
If is a unital Banach algebra and is a nonzero complex homomorphism on , then .
Let with . If with then , and by Theorem 2 we have . Then,
hence , i.e. . This shows that . ∎
The Gleason-Kahane-Zelazko theorem22 2 Walter Rudin, Functional Analysis, second ed., p. 251, Theorem 10.9 states that if is a unital Banach algebra and is a linear functional on satisfying and for , then is a complex homomorphism.
In Theorem 2 we proved that a perturbation of the identity element remains in the multiplicative group. We now prove that for any element of the multiplicative group, a sufficiently small perturbation of this element remains in the multiplicative group, i.e. that the multiplicative group is an open set.33 3 Walter Rudin, Functional Analysis, second ed., p. 253, Theorem 10.11.
If is a unital Banach algebra, , , and , then , and
If is a unital Banach algebra and , the spectrum of is the set
and the spectral radius of is
If then and so by Theorem 2,
so . Therefore,
For , we define the resolvent of by
The following is the resolvent identity.
Theorem 5 (Resolvent identity).
If is a unital Banach algebra, , and , then
The resolvent set of is . The following theorem implies that the resolvent set of is open, and therefore that the spectrum of is closed. Because , it follows that is a compact set.
If is a unital Banach algebra, , and , then implies that .
Moreover, is nonempty, and although may be strictly less than , the spectral radius is equal to a limit of norms.44 4 Gert K. Pedersen, Analysis Now, revised printing, p. 131, Theorem 4.1.13. This proof does not use the holomorphic functional calculus, while Walter Rudin, Functional Analysis, second ed., p. 253, Theorem 10.13 does. That is, to read Pedersen’s proof one does not have to make sense of integrals of functions taking values in a Banach space, while to read Rudin’s one does.
If is a unital Banach algebra and , then is nonempty, and
Using the fact that the spectrum of any element of a unital Banach algebra is nonempty, one can prove the Gelfand-Mazur theorem, which states that if is a unital Banach algebra for which every nonzero element is invertible, then there is an isometric algebra isomorphism .55 5 Walter Rudin, Functional Analysis, second ed., p. 255, Theorem 10.14.
The following theorem shows that is a continuous function . From this and the resolvent identity it follows that for any ,
If is a unital Banach algebra and , then is a continuous function .
From Theorem 5, if then
where we write . As
The following theorem tells us that if the spectrum of an element of a unital Banach algebra is contained in an open set, then the spectrum of a sufficiently small perturbation of that element is contained in the same open set.66 6 Walter Rudin, Functional Analysis, second ed., p. 257, Theorem 10.20.
If is a unital Banach algebra, , and is an open subset of containing , then there is some such that implies that .
If , then
Hence . By Theorem 8, is a continuous function . It follows that there is some such that implies that . If and , then
which implies that . From this we obtain that
which means that . We have shown that if then . ∎
The following theorem gives conditions under which we can evaluate a holomorphic function at an element of a unital Banach algebra, and shows that the image of the spectrum is contained in the spectrum of the image.
If is a unital Banach algebra, , , and is holomorphic on a domain that contains the closed disc , then
and if then .
Because the radius of convergence of is , it follows that the above sums tend to as , showing that . Define
But if then
and if then
and as has radius of convergence this tends to as . Therefore
If then , and as we have written as a product of and another element of , it follows that , and thus . ∎
If is a unital Banach algebra, , and , then