Banach algebras abstract the properties of closed algebras of operators on Banach spaces. Many basic properties of such operators have elegant proofs in the framework of Banach algebras, and Banach algebras also naturally appear in areas of mathematics like harmonic analysis, where one writes down Banach algebras generalizing the group algebra to study topological groups.
Today we will develop some of the basic theory of Banach algebras, our goal being to discuss the Gelfand representation of a commutative Banach algebra and the fact that, for commutative C*-algebras, this representation is an isometric isomorphism. This implies in particular a spectral theorem for self-adjoint operators on a Hilbert space.
This material can be found in many sources; I am working from Dales, Aiena, Eschmeier, Laursen and Willis’ Introduction to Banach Algebras, Operators, and Harmonic Analysis.
Below all vector spaces are over , all algebras are unital, and all algebra homomorphisms preserve units unless otherwise stated. In the context of Banach algebras, the last two assumptions are not standard, but in practice non-unital Banach algebras are studied by adjoining units first, so we do not lose much generality.
Definitions and examples
A Banach algebra is a Banach space which is also an associative algebra (with the same addition) such that the multiplication satisfies and . Abstractly, it is a monoid object in . A morphism of Banach algebras is a weak contraction which is also an algebra homomorphism. Note that since we want morphisms to preserve units, any morphism has norm exactly .
(Note also that the zero ring cannot be a Banach algebra with these axioms since and is incompatible with .)
Example. The fundamental example is the algebra bounded linear operators on a Banach space with the operator norm.
Example. Let be a bounded linear operator on a Banach space. Then the closure of the algebra is a commutative Banach algebra. The application of Banach algebra techniques to spectral theory proceeds by studying algebras like this.
Example. Let be a collection of bounded linear operators on a Banach space . Their commutant is an intersection of closed subalgebras of , hence also a Banach algebra. If is thought of as a module over the algebra generated by , then consists of the endomorphisms of this module.
Example. Let be a topological space and consider again the space of bounded continuous functions . This space is a Banach space when equipped with the sup norm, and moreover pointwise multiplication makes it a Banach algebra. If is discrete, this is just . If is compact, this is just .
Example. Let be a locally compact Hausdorff space which is not compact. We denote by the space of continuous functions which vanish at infinity in the sense that for any there exists a compact set such that for ; these are precisely the functions vanishing at the point at infinity in the one-point compactification of . This condition is preserved by uniform limits, so is a Banach space when equipped with the sup norm. When equipped with pointwise products, it is a non-unital Banach algebra (the identity cannot vanish at infinity). Adjoining a unit corresponds to passing to the one-point compactification.
Example. Let be a small category. Recall that the category ring is the free abelian group on the morphisms of , with the product of two morphisms being their composition if defined and zero otherwise (and recall that it is unital if and only if has finitely many objects.) Since it is in particular a free abelian group on a set, we can tensor it with and equip it with the norm; the resulting completion is non-unital if and only if has infinitely many objects. Even if has finitely many objects, in which case the unit of is the sum of the identity morphisms in , that unit satisfies , which is usually not equal to .
Thinking of in terms of functions , the product is the convolution
and so is called the convolution algebra of . This definition is usually only stated in the special case that is a group (that is, a one-object small category with all morphisms invertible), in which case it generalizes the group algebra, although it is sometimes also given when is a groupoid.
Groupoids naturally arise when some space of interest is quotiented by some equivalence relation or group action. In such situations it often happens that the quotient as a space is very badly behaved, and Alain Connes’ approach to understanding this situation via noncommutative geometry is to replace the quotient by the (noncommutative space corresponding to the) convolution algebra of the corresponding groupoid.
Sub-example. Taking to be the groupoid of objects, all of which are isomorphic via a unique isomorphism, we recover (with norm given by the sum of the absolute values of the entries). This is more or less how Heisenberg discovered matrix mechanics: rather than thinking in terms of vector spaces he thought in terms of transitions between energy levels, which form the morphisms in a groupoid.
The category above is equivalent but not isomorphic to the category with one object and a single morphism. The corresponding algebra is just , which is similarly Morita equivalent but not isomorphic to .
Example. Let be a locally compact group (e.g. ). has a nice left-invariant measure called Haar measure (e.g. Lebesgue measure) which is unique up to scaling; if is compact we can normalize it so that to get normalized Haar measure. The Banach space can also be equipped with a convolution product, which for groups can be written
When is a group with the discrete topology, counting measure is a Haar measure and we recover . Like , typically is not unital (for example when ).
Compact Hausdorff spaces
Theorem: The functors define a contravariant equivalence of categories from the category of compact Hausdorff spaces to a subcategory of the category of -algebras (where is given the Zariski topology).
Banach algebras are an abstraction of closed subalgebras of the algebra of bounded operators on a Banach space in the same way that groups are an abstraction of subgroups of the symmetric group on a set. Accordingly, 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 et al.
Basic properties assuming the spectrum is non-empty
The following is fundamental.
Black Box #1: The spectrum is non-empty.
The idea of the proof is that if were empty, then the resolvent would be a non-constant “bounded holomorphic function,” which contradicts Liouville’s theorem. This proof actually works with no modifications for complex matrices and shows that they have eigenvalues without going through either characteristic polynomials or the fundamental theorem of algebra, and in fact this is one way to prove the fundamental theorem of algebra. In general we need to develop some complex analysis for Banach space-valued holomorphic functions, which we defer to a later post.
For now, it it is more important to see what can be done with this result.
Corollary (Gelfand-Mazur): Any Banach algebra which is a division algebra is isometrically isomorphic to .
Proof. Let be such a Banach algebra and let be nonzero. is non-empty, so let be such that is not invertible. Since is nonzero, it must be invertible, so , hence is not invertible. But then it must be equal to , so .
Corollary: Let be a commutative Banach algebra. Then if and only if there exists a morphism such that .
Proof. : if there exists such an , then lies in a maximal ideal, so cannot be invertible. (This argument works in any algebra over .)
: if , then is not invertible, hence is contained in a maximal ideal , which is already closed. Since by Gelfand-Mazur, the quotient map is the desired .
(Note that we make essential use of the axiom of choice here.)
A digression into quantum mechanics
The non-emptiness of the spectrum, besides leading to the above results, also has the following corollary, which is apparently due to Wielandt.
Proposition: The Weyl algebra does not embed into a Banach algebra.
Proof. Suppose to the contrary that are elements of a Banach algebra satisfying . We make the elementary observation that , from which it follows that
But we know that , so must be invariant under addition by . Since is non-empty, it must contain arbitrarily large elements, and this contradicts compactness.
This result is not needed for what follows, but it does imply that it is not possible to describe the position and momentum operators as elements of a Banach algebra. Equivalently, it is not possible to represent position and momentum as bounded operators on a Banach space. In their usual representation on , both position and momentum are unbounded (discontinuous) and densely defined (can only be applied to certain dense subspaces of ).
Fortunately, there is a way out: instead of dealing with position and momentum directly, we can deal with the strongly continuous one-parameter unitary groups they generate via Stone’s theorem (see also the Stone-von Neumann theorem). Indeed, these unitary groups are defined on by
for position and (up to normalization)
for momentum, both of which are defined everywhere on .
The Gelfand representation
Recall that for any commutative ring we can define a homomorphism
where the product runs over all maximal ideals of and is the image of in . This represents as a “ring of functions” on , although ideally all of the residue fields should be the same for this interpretation to really make sense, and the kernel of this homomorphism is the Jacobson radical .
We now know that if is a Banach algebra, we always have , so the above homomorphism lands in the space of functions . It’s natural to want to equip , the spectrum, Gelfand spectrum, or Gelfand space of , with a topology such that the corresponding functions are all continuous. This is usually done by identifying an element of with the corresponding map , which is in particular a continuous linear functional, hence naturally sits as a subspace of . The initial topology on making all evaluations against elements of continuous is the weak-* topology, and the topology we want on is the subspace topology with respect to this topology.
(An equivalent way to define the weak-* topology, which makes sense for replaced by any topological vector space , is to think of as a subspace of the space of functions , or equivalently a subspace of the product of copies of ; then the weak-* topology is the subspace topology with respect to the product topology on this product.)
Since morphisms have norm , in fact lies in the unit ball of . We now need the following result, which shows that the weak-* topology is very different from the norm topology on .
Theorem (Banach-Alaoglu): let be a normed space. Then the unit ball of is compact in the weak-* topology.
Proof. Thinking of as a subspace of with the product topology, the unit ball consists of linear functionals which send the unit ball of to the unit ball (disk) , hence can be thought of as a subspace of with the product topology. This is compact by Tychonoff’s theorem. The unit ball of is a closed subspace of (since it is defined by the additional requirement that it consists of linear functions, which impose a collection of closed conditions), hence is also compact.
(Note that is compact Hausdorff, so we only need Tychonoff’s theorem for compact Hausdorff spaces here, hence we only need the ultrafilter lemma and not the full strength of the axiom of choice. In fact Banach-Alaoglu is equivalent to the ultrafilter lemma.)
Corollary: For a commutative Banach algebra, is compact Hausdorff in the weak-* topology.
It follows that the homomorphism above, applied to a commutative Banach algebra, gives us its Gelfand representation
Since is compact Hausdorff, the space of continuous functions on it is itself a Banach algebra with the sup norm. We record the following properties of the Gelfand representation to summarize what we have already proven.
- has range the spectrum of .
- Consequently is the spectral radius .
- Consequently the Gelfand representation is a weak contraction (hence a morphism of Banach algebras).
- The kernel of the Gelfand representation is . It is also the set of elements of with spectral radius zero (the quasi-nilpotent elements).
- Consequently the Gelfand representation is faithful if and only if is semiprimitive, if and only if has no nonzero quasi-nilpotents.
(Some sources use “semisimple” to mean semiprimitive. The people who use this convention use “Artinian semisimple” to mean semisimple.)
Commutative semiprimitive Banach algebras therefore have a relatively concrete description as algebras of continuous functions on compact Hausdorff spaces; note, however, that even if is semiprimitive we are not guaranteed that the spectral radius coincides with the norm on . For an important class of commutative Banach algebras which we now introduce, we will be able to show that the Gelfand representation is an isometric isomorphism.
For a Hilbert space, the Banach algebra of bounded linear operators on crucially has extra structure not present in a general Banach algebra given by the adjoint. In abstracting this structure, we are led to the following sequence of definitions.
A *-ring (or involutive ring, or ring with involution) is a ring equipped with an anti-homomorphism which squares to the identity. Explicitly, this is a map satisfying
Equivalently, a *-ring is a small -enriched dagger category, so I am going to call them -rings and use for the involution even though “dagger ring” appears to have some unrelated meaning in mathematics. A morphism of -rings is a ring homomorphism preserving the involution.
Example. Any commutative ring is a -ring with trivial involution.
Example. is a -ring with the involution being complex conjugation.
Example. The quaternions are a -ring with the involution sending to .
Example. Let be a small dagger category. The category ring is a -ring with the involution extending the involution on . In particular, a groupoid is a dagger category with the involution being inverse, so we get an example for any groupoid and in particular any group.
Example. Let be a commutative ring. The ring of matrices over is a -ring with the involution being transpose. (This is more or less a special case of the above applied to the groupoid of equivalent objects after tensoring with .)
A *-algebra over a commutative *-ring is an algebra over in the ordinary sense which is also equipped with an involution making it a *-ring compatible with the one on . Equivalently, it is a *-ring together with a morphism of *-rings. Again, I will call these -algebras. A morphism of -algebras is an -linear morphism of -rings.
Example. Let be a -ring which is also an algebra over (with the involution restricted to being trivial). Then the tensor product is a complex -algebra (that is, a -algebra over as a -ring) with the involution being complex conjugation on and the existing involution on .
Letting be the category ring of a dagger category we get the complex -algebras .
Example. The algebra is a complex -algebra with the involution being adjoint, as is any -closed subalgebra.
We will primarily be interested in complex -algebras, which are -algebras over regarded as a -ring using complex conjugation. In the commutative case, this is more or less a more convenient way to study real algebras.
Theorem: The functor sending a commutative complex -algebra to its subalgebra of self-adjoint elements is an equivalence of categories to the category of commutative -algebras.
Proof. The inverse of this functor sends an algebra to the tensor product with involution given by complex conjugation. That this is the inverse follows from the fact that for any element of a commutative complex -algebra we can write
which is precisely an element of tensor the self-adjoint subalgebra of ; moreover, morphisms out of are uniquely and freely determined by what they do to self-adjoint elements by the above.
Example. Let be the algebra of Laurent polynomials with involution given by . Identifying with the space of Fourier polynomials , we may identify the self-adjoint subalgebra with the space of trigonometric polynomials .
We may think of the above as being essentially about the Galois theory of the Galois extension ; more generally, if is any Galois extension, then there is a functor
and we may promote this to an equivalence of categories by attaching to the data of the action of the Galois group on the left factor; taking the -invariant subalgebra then gives the inverse functor. None of this requires that the -algebras or -algebras in question are commutative.
However, noncommutative complex -algebras do not fit into this formalism since the involution reverses the order of multiplication. In particular, the self-adjoint elements no longer form a subalgebra; if are self-adjoint but don’t commute then
The self-adjoint elements are instead closed under the anticommutator
and axiomatizing this observation leads to the theory of Jordan algebras. However, we won’t be concerned much with anticommutators; if are observables in a quantum system, we generally prefer to look at the observable , for example when is position and is momentum, in view of Noether’s theorem.
We are now ready for the following definition axiomatizing the properties of closed -closed subalgebras of . A C*-algebra is a Banach algebra equipped with an involution making it a complex -algebra and also satisfying the B*-identity
(See Wikipedia for the history here.) A morphism of C*-algebras is a morphism of Banach algebras which respects the involution.
(It is not obvious that the B*-identity is either useful or enough to capture everything we want. Happily, both are true.)
Proposition: Let be a Hilbert space. Then the algebra of bounded linear operators is a C*-algebra with involution the adjoint.
Proof. Let be such an operator. Then
and the conclusion follows.
Corollary: Any closed -closed subalgebra of is also a C*-algebra.
Example. Let be a bounded linear operator. Then generate a -closed subalgebra of whose closure is also -closed (this follows from the fact that taking adjoints preserves norms, hence is itself continuous with norm ), hence gives a C*-algebra. This C*-algebra is commutative if and only if is normal.
Example. Let be a collection of bounded linear operators on a Hilbert space which is -closed. Then their commutant is also -closed, hence is a C*-algebra. These C*-algebras are precisely the von Neumann algebras.
Example. If is a compact Hausdorff space then is a commutative C*-algebra with the involution being complex conjugation. It may be regarded as a space of operators acting by multiplication on if is equipped with a suitably nice Borel measure, although I am not sure of the precise hypotheses we need for the operator norm to match the sup norm. At a minimum the measure of every open set must be positive.
We now need the following.
Black Box #2 (Beurling-Gelfand spectral radius formula): Let be an element of a Banach algebra . Then the spectral radius satisfies
This is a surprising result in light of the fact that the LHS depends only on the algebra structure of while the RHS is defined in terms of the norm on . The crux of the proof is the following version of the Cauchy integral formula, which we will not prove here:
for all . If , then it follows that
hence that for all . Since we already know that , the conclusion follows. Again, we defer discussion of the relevant complex analysis to a later post.
Corollary: Let be a self-adjoint element of a C*-algebra. Then .
Proof. The B*-identity gives , and iterating this gives
for all . Taking -th roots of both sides and taking limits, the conclusion follows by the spectral radius formula.
Corollary: Let be an element of a C*-algebra . Then . Consequently, .
Proof. The B*-identity and the above gives since is self-adjoint. Now, , and since is invertible if and only if is invertible, it follows that , hence , and the conclusion follows.
So the norm on a C*-algebra is already uniquely determined by its complex -structure. Moreover:
Corollary: Let be C*-algebras. Any complex -algebra morphism is a weak contraction; consequently, it is already a morphism of C*-algebras. Equivalently, the category of C*-algebras is a full subcategory of the category of complex -algebras.
Proof. If is invertible in then is invertible in . Consequently, , so . By the above, it then follows that .
Before we proceed, we will also need the following.
Lemma: Let be a self-adjoint element of a C*-algebra . Then has real spectrum.
This sum converges absolutely for all and we have . Standard analytic arguments show that
and we know that by the above. It follows that if is any morphism of Banach algebras, it is in particular a weak contraction, so . But and ; combining these two conditions we conclude that has zero imaginary part.
We now have enough to prove the following remarkable result.
Theorem (commutative Gelfand-Naimark): Let be a commutative C*-algebra. Then the Gelfand representation is an isomorphism of C*-algebras.
In particular, commutative C*-algebras are semiprimitive.
Proof. It remains to show that 1) the Gelfand representation respects , 2) the Gelfand representation is an isometry (hence ) and in particular injective, 3) the Gelfand representation is surjective.
1): Let where are self-adjoint (recall that every element of is uniquely of this form). If is a morphism, then are real by the lemma above, hence
2): This follows from and 1), since the Gelfand representation preserves spectral radius. If it follows now that , hence , hence the Gelfand representation is injective.
3): By 2), may be identified with a closed subalgebra of with norm the induced norm which is closed under complex conjugation, and by definition of the maximal spectrum it separates points. By the complex Stone-Weierstrass theorem, it then follows that .
Remember that when we exhibited a contravariant equivalence between compact Hausdorff spaces and certain -algebras we did not specify what algebras arose in this way. The Gelfand-Naimark theorem supplies the answer: they are precisely the self-adjoint subalgebras of commutative C*-algebras. Combined with results we already know about this equivalence, we conclude the following.
Gelfand duality: The functors form a contravariant equivalence between commutative C*-algebras and compact Hausdorff spaces.
Corollary: Let be a topological space. Then is the Stone–Cech compactification of .
Gelfand duality is a very suggestive theorem: it suggests that doing topology with compact Hausdorff spaces is equivalent to doing algebra with commutative C*-algebras. The idea that noncommutative C*-algebras therefore constitute a generalization of compact Hausdorff spaces is the basis of the noncommutative topology program.
But even without getting into such issues, Gelfand duality also has relatively concrete applications. For example, it implies a weak form of the spectral theorem: if is a normal operator on a Hilbert space , then considering the commutative C*-algebra generated by it follows that is isomorphic, in an appropriate sense, to a multiplication operator
where is a continuous complex-valued function on a compact Hausdorff space (namely the image of in ). Ideally we would want to be an element of and we would want the identification between and to be unitary, and it seems that we need to find an appropriate Borel measure on to say this. However, even without such a measure we still see that is diagonalizable in a suitable sense.
If is finite-dimensional, then so is , and this is possible if and only if is finite with the discrete topology; call its points . Note that must be the number of distinct eigenvalues of and the points may be identified with the set (not multiset) of eigenvalues of . Moreover, contains distinguished idempotents , the indicator functions of the points . These elements are self-adjoint idempotents in , so they must in fact be projections, and they are precisely the projections onto the eigenspaces of . So the spectral theorem provided by Gelfand duality really does recover the ordinary finite-dimensional spectral theorem.
Another application is as follows. Let be a normal element of a C*-algebra . The closure of defines a commutative C*-subalgebra of (the minimal such thing). By Gelfand duality, . Now, the latter space is closed under composition by any continuous function , Hence if is any such function, we can define a continuous functional calculus which allows us to apply continuous functions to normal elements of and to get other elements of . For example, we can define the absolute value (not to be confused with the norm ) by taking the pointwise absolute value in .
only needs to be defined on the spectrum of in order for to be well-defined. Thus if is self-adjoint then we can make sense of for any continuous . If is self-adjoint and has positive spectrum then we can make sense of, for example, and .
As a more specific application (although I do not think I need the full power of the continuous functional calculus to conclude this), recall that a one-parameter subgroup of a topological group is a continuous group homomorphism .
Theorem: The map assigning an element of a C*-algebra the function is a bijection from self-adjoint elements of to norm-continuous one-parameter subgroups of the unitary group of .
Proof. If is self-adjoint, lands in the commutative C*-subalgebra generated by , and any one-parameter subgroup of takes values in a commutative C*-subalgebra, so we may assume without loss of generality that is commutative. If is self-adjoint, then , thought of as a one-parameter family of elements of , is a one-parameter subgroup by the boundedness of , and taking adjoints shows that it is unitary. Conversely, if is a one-parameter subgroup of , then for sufficiently small , as a function on , will only take values in a ball of small radius around by norm-continuity. On a sufficiently small such ball we can define a branch of the logarithm, and then is a one-parameter family of functions such that . Since is continuous in , it follows that for some , and by unitarity is self-adjoint.
The above theorem describes the exponential map from the unitary Lie algebra
of skew-adjoint elements of (which are precisely the elements where is self-adjoint) to the unitary group . However, it is stated for self-adjoint elements so that it becomes a version of Noether’s theorem for C*-algebras, with self-adjoint elements of a C*-algebra taking the role of observables of some quantum system (the noncommutative space associated to ) and one-parameter subgroups of taking the role of one-parameter groups of symmetries of the system.
Note that already to talk about position and momentum we needed unbounded operators which can’t embed into a Banach algebra or strongly continuous (not norm-continuous) one-parameter subgroups, so this is an incomplete discussion.