Gelfand Representation

Mathematical representation in functional analysis

In the grand tapestry of mathematics, specifically within the esoteric realm of functional analysis, the Gelfand representation (a concept graciously bequeathed by I. M. Gelfand, who, one presumes, found these truths rather self-evident) manifests itself in one of two distinct, yet profoundly interconnected, forms. It's almost as if the universe decided to offer a convenient translation service for the abstract.

Firstly, it provides a rather elegant, if somewhat predictable, method for representing commutative Banach algebras as algebras composed of continuous functions. For those who enjoy seeing the abstract made concrete, this is quite the spectacle. It transforms an algebraic structure, defined by operations and a norm, into a space where elements are simply functions, and operations are performed pointwise – a much more intuitive landscape for many. This particular aspect of the Gelfand representation offers a far-reaching generalization of concepts familiar from classical analysis, such as the ubiquitous Fourier transform of an integrable function. One might consider the Fourier transform not as a unique phenomenon, but rather as an early, specific instance of this broader Gelfand principle, a testament to its pervasive nature.

Secondly, and arguably more profoundly, for the more structured and well-behaved commutative C*-algebras, this representation elevates itself to the status of an isometric isomorphism. This isn't just a mere mapping; it's a perfect, structure-preserving, distance-preserving equivalence. It means that the abstract C*-algebra is, for all intents and purposes, identical to an algebra of continuous functions. This latter case, often encapsulated within the broader Gelfand–Naimark representation theorem, serves as a foundational pillar in the development of spectral theory for normal operators. It offers a powerful and comprehensive generalization of the rather mundane algebraic task of diagonalizing a normal matrix in finite dimensions, extending this crucial concept to the infinitely more complex world of infinite-dimensional spaces. The implications for understanding the intrinsic properties of operators are, frankly, immense, even if the journey to appreciate them is often paved with existential dread.

Historical remarks

One of Gelfand's initial forays into the practical utility of this representation – and one which, predictably, spurred a significant surge in the study of Banach algebras (because humans, bless their hearts, often require immediate application to validate abstract beauty) – was to furnish a considerably more concise and conceptually clear proof of a celebrated lemma. This lemma, originally formulated by the rather prolific Norbert Wiener, characterized the elements within the group algebras L¹(R) and ℓ¹(Z) whose translates managed to span dense subspaces within their respective algebraic structures. Wiener's original work, while groundbreaking, involved intricate techniques. Gelfand's application of his representation theorem provided a proof that, in retrospect, seemed almost inevitable, transforming a complex analytical problem into a more transparent algebraic one. It demonstrated that by shifting perspective through the Gelfand transform, previously intractable problems could yield to elegant and insightful solutions, thereby solidifying the practical relevance of these abstract algebraic structures.

The model algebra

To truly appreciate the Gelfand representation, one must first understand its ideal form, the very blueprint it strives to emulate. For any given locally compact Hausdorff topological space X (conditions that, as one might observe, are rather specific but entirely necessary for well-behaved spaces), the space denoted C₀(X) – comprising all continuous complex-valued functions on X that vanish at infinity – naturally forms a commutative C*-algebra. This isn't a mere coincidence; it's a consequence of its inherent structure:

The algebra structure over the complex numbers is established through the most straightforward means imaginable: pointwise operations of addition and multiplication. Each function is treated independently at every point, making the algebra's operations utterly transparent.
The involution (that critical * operation for C*-algebras) is simply pointwise complex conjugation. Again, a natural and intuitive operation that preserves the essential structure when dealing with complex-valued functions.
The norm, which dictates the "size" of these functions, is the uniform norm, also known as the supremum norm. It measures the maximum absolute value a function attains, ensuring that convergence in this space corresponds to uniform convergence, a powerful analytical property.

The importance of X being both locally compact and Hausdorff cannot be overstated, though one often finds these conditions taken for granted. These topological properties ensure that X is a completely regular space (specifically, a Tychonoff space). This regularity is crucial because it allows for a remarkable correspondence: every closed subset of X can be precisely defined as the common zero set of some family of continuous complex-valued functions on X. This property, often called function space separation, means that the topological structure of X can be entirely recovered from the algebraic and analytical properties of C₀(X). In essence, the functions encode the space itself, a rather profound observation that lays the groundwork for the Gelfand representation's power.

It is also worth noting, for those who appreciate such details, that C₀(X) possesses a multiplicative identity element, making it a unital algebra, if and only if the underlying space X is a compact space. In this specific scenario, C₀(X) simplifies to C(X), which is simply the algebra of all continuous complex-valued functions on X, as the condition of "vanishing at infinity" becomes trivial when there is no "infinity" to vanish towards.

Gelfand representation of a commutative Banach algebra

Now, let's consider a more general specimen: let $A$ be an arbitrary commutative Banach algebra, defined over the field $\mathbb{C}$ of complex numbers. The fundamental building block of the Gelfand representation is the concept of a "character." A non-zero algebra homomorphism, denoted $\Phi \colon A\to \mathbb{C}$ , is precisely what we call a character of $A$ . In simpler terms, it's a linear map that preserves multiplication and maps elements of the algebra to complex scalars. The collection of all such characters of $A$ is denoted by $\Phi _{A}$ .

One might initially wonder about the behavior of these characters. It's a non-trivial, yet ultimately predictable, fact that every character on $A$ is automatically continuous. This is a powerful result, as it means we don't need to impose continuity as an additional condition; it's inherent to the algebraic structure. Consequently, $\Phi _{A}$ naturally forms a subset of $A^{*}$ , the topological dual space of continuous linear functionals on $A$ . Furthermore, when endowed with the relative weak-* topology (which, for those unfamiliar, is the coarsest topology where all evaluation maps remain continuous), $\Phi _{A}$ transforms into a locally compact and Hausdorff topological space. This topological elegance is not accidental; it’s a direct consequence of the profound Banach–Alaoglu theorem, which guarantees the weak-* compactness of the unit ball in a dual space, and from which the properties of $\Phi _{A}$ can be derived. A subtle but important point is that $\Phi _{A}$ achieves full compactness (within this topology) if and only if the algebra $A$ happens to possess an identity element. [1]

With this space of characters, $\Phi _{A}$ , firmly established, we can define the core of the representation: the Gelfand transform. For any element $a\in A$ , we define a function $\widehat{a}:\Phi _{A}\to {\mathbb {C} }$ by the deceptively simple rule $\widehat{a}(\phi )=\phi (a)$ . This function, $\widehat{a}$ , is known as the Gelfand transform of $a$ . The very definition of $\Phi _{A}$ and the topology imposed upon it ensure that $\widehat{a}$ is not only continuous but also vanishes at infinity [2].

The mapping $a\mapsto {\widehat {a}}$ itself is the Gelfand representation of $A$ . This map constitutes a norm-decreasing, unit-preserving algebra homomorphism from $A$ to $C_{0}(\Phi _{A})$ . The "norm-decreasing" property means that the norm of the transformed element is less than or equal to the norm of the original element, ensuring a certain boundedness. "Unit-preserving" means that if $A$ has an identity, it maps to the identity function in $C_{0}(\Phi _{A})$ . However, it's crucial to understand that, in its most general form, this representation is neither guaranteed to be injective (meaning distinct elements in $A$ might map to the same function) nor surjective (meaning not every function in $C_{0}(\Phi _{A})$ necessarily corresponds to an element in $A$ ).

The question of injectivity leads to deeper insights into the structure of $A$ . In the scenario where $A$ possesses an identity element, there exists a profound bijection between $\Phi _{A}$ and the set of maximal ideals in $A$ . This connection is a direct consequence of the powerful Gelfand–Mazur theorem, which states that any Banach algebra that is also a field must be isomorphic to the complex numbers. As a result of this deep algebraic-topological link, the kernel of the Gelfand representation $A\to C_{0}(\Phi _{A})$ can be precisely identified with the Jacobson radical of $A$ . This is a critical observation: the Jacobson radical essentially captures all the "non-semisimple" parts of the algebra, the elements that prevent a faithful representation. Thus, the Gelfand representation achieves injectivity if and only if $A$ is (Jacobson) semisimple, meaning its Jacobson radical is trivial.

Examples

To truly grasp the utility of the Gelfand representation, one must look at its concrete manifestations. These examples serve as a kind of Rosetta Stone, translating abstract theory into familiar analytical tools.

Consider the Banach space $A=L^{1}(\mathbb{R})$ , which, under the operation of convolution, forms a Banach algebra. This is, in fact, a classic instance of a group algebra of a locally compact group. In this specific context, the space of characters $\Phi _{A}$ turns out to be homeomorphic to $\mathbb{R}$ itself. More strikingly, the Gelfand transform of an element $f\in L^{1}(\mathbb{R})$ is precisely the well-known Fourier transform, typically denoted $\tilde{f}$ . It's almost as if the Fourier transform was merely waiting for Gelfand to generalize its underlying mechanism. Similarly, if one takes $A=L^{1}(\mathbb{R}_{+})$ , representing the group algebra of the multiplicative reals (positive real numbers under multiplication), the Gelfand transform yields the equally important Mellin transform. These examples underscore how the Gelfand representation provides a unifying framework for various integral transforms that are fundamental to analysis.

Another illuminating example arises with $A=\ell ^{\infty}$ , the Banach algebra of bounded sequences. Here, the representation space $\Phi _{A}$ is revealed to be the Stone–Čech compactification $\beta \mathbb{N}$ of the natural numbers. This is a fascinating result, as $\beta \mathbb{N}$ is a much "larger" and more complex space than $\mathbb{N}$ itself, reflecting the richness of the characters on $\ell ^{\infty}$ . More generally, if $X$ is any completely regular Hausdorff space, then the representation space of the Banach algebra of bounded continuous functions on $X$ is precisely the Stone–Čech compactification of $X$ [2]. These connections reveal a deep interplay between abstract algebra and advanced topology.

The C*-algebra case

As a motivating preamble, let us revisit the "model" algebra, the space $A = C_{0}(X)$ , consisting of continuous functions vanishing at infinity on a locally compact Hausdorff space $X$ . For any point $x$ in $X$ , one can define a functional $\varphi _{x}\in A^{*}$ that simply evaluates a function at that point: $\varphi _{x}(f)=f(x)$ . One might observe, with a nod to the obvious, that $\varphi _{x}$ is indeed a character on $A$ , as it is linear and preserves multiplication. What is more profound, and perhaps less immediately obvious, is that all characters of $A$ are precisely of this form. This means there's a perfect one-to-one correspondence between the points of the topological space $X$ and the characters of the algebra $C_{0}(X)$ . A more precise and rigorous analysis reveals that we can identify $\Phi _{A}$ with $X$ , not merely as sets, but as isomorphic topological spaces. Consequently, for this "model" algebra, the Gelfand representation becomes an actual isomorphism: $C_{0}(X)\to C_{0}(\Phi _{A})$ . This isn't just a mapping; it's an exact structural equivalence, a perfect mirror image. This foundational insight sets the stage for the full power of the Gelfand–Naimark theorem for C*-algebras.

The spectrum of a commutative C*-algebra

The term "spectrum" in functional analysis is, regrettably, somewhat overloaded, a common affliction in mathematics. One must be precise. The spectrum, or more specifically the Gelfand space, of a commutative C*-algebra $A$ , often denoted $\widehat{A}$ , consists of the set of all non-zero -homomorphisms from $A$ to the complex numbers. These elements are, once again, referred to as characters on $A$ . It's a rather useful fact, though not immediately apparent, that any algebra homomorphism from a C-algebra $A$ to the complex numbers is automatically a *-homomorphism (i.e., it also preserves the involution operation) and is automatically continuous. Thus, this definition of 'character' aligns perfectly with the one previously introduced for general Banach algebras, simplifying things marginally for the C*-algebra case.

Crucially, the spectrum of a commutative C*-algebra is not just a set; it's endowed with a natural topology, specifically the weak-* topology inherited from the dual space. This makes $\widehat{A}$ a locally compact Hausdorff space. In the special case where the C*-algebra $A$ is unital (i.e., it possesses a multiplicative identity element 1), all characters $f$ must be unital, meaning $f(1)$ must be the complex number one. This condition naturally excludes the zero homomorphism. In this scenario, $\widehat{A}$ is not merely locally compact but fully compact. Conversely, for a non-unital C*-algebra, the weak-* closure of $\widehat{A}$ is $\widehat{A} \cup \{0\}$ , where $0$ represents the zero homomorphism. The removal of this single point (the zero homomorphism) from a compact Hausdorff space then yields a locally compact Hausdorff space.

Now, to address the aforementioned "overloaded word": "spectrum" also refers to the spectrum $\sigma(x)$ of an individual element $x$ within an algebra with a unit 1. This is defined as the set of all complex numbers $r$ for which $x - r1$ is not invertible in $A$ . For unital C*-algebras, these two notions of "spectrum" are intimately connected, as one might expect from a well-designed mathematical theory. Specifically, $\sigma(x)$ is precisely the set of complex numbers $f(x)$ where $f$ ranges over the Gelfand space $\widehat{A}$ . This connection, coupled with the spectral radius formula (which states that the spectral radius of a normal element equals its norm), reveals that $\widehat{A}$ is a subset of the unit ball of $A^{*}$ and, as such, can be equipped with the relative weak-* topology. This topology is, in essence, the topology of pointwise convergence. A net $\{f_k\}_k$ of elements within the spectrum of $A$ converges to $f$ if and only if for each $x$ in $A$ , the net of complex numbers $\{f_k(x)\}_k$ converges to $f(x)$ .

Should $A$ be a separable C*-algebra (meaning it contains a countable dense subset, a property that often simplifies analysis), the weak-* topology becomes metrizable on bounded subsets. This is a significant practical advantage, as it means the spectrum of a separable commutative C*-algebra $A$ can be regarded as a metric space, allowing its topology to be characterized through the more familiar notion of convergence of sequences, rather than just nets. Equivalently, and perhaps more succinctly, the spectrum $\sigma(x)$ of an element $x$ is simply the range of its Gelfand transform $\gamma(x)$ .

Statement of the commutative Gelfand–Naimark theorem

And now, for the main event, the culmination of these observations. Let $A$ be a commutative C*-algebra, and let $X$ denote its spectrum $\widehat{A}$ (the Gelfand space, as discussed above). Let $\gamma:A\to C_{0}(X)$ be the Gelfand representation, meticulously defined in the preceding sections.

Theorem. The Gelfand map $\gamma$ is an isometric *-isomorphism from $A$ onto $C_{0}(X)$ .

This theorem, a cornerstone of functional analysis, tells us something profound (see the Arveson reference below). It declares that every abstract commutative C*-algebra is, up to an isometric -isomorphism, precisely an algebra of continuous complex-valued functions vanishing at infinity on some locally compact Hausdorff space. "Isometric" means it preserves norms (distances), and "-isomorphism" means it preserves the algebraic operations (addition, multiplication), scalar multiplication, and the involution (the * operation). In essence, these two seemingly disparate mathematical objects are structurally identical.

The spectrum of a commutative C*-algebra can also be conceptualized as the set of all maximal ideals $m$ of $A$ , endowed with the hull-kernel topology. This perspective, already hinted at in the general commutative Banach algebra case, becomes even more powerful here. For any such maximal ideal $m$ , the quotient algebra $A/m$ is, by virtue of the Gelfand–Mazur theorem, one-dimensional, and thus isomorphic to $\mathbb{C}$ . Therefore, any element $a$ in $A$ naturally gives rise to a complex-valued function on this space of maximal ideals. This algebraic construction perfectly aligns with the character-based definition of the Gelfand space.

For C*-algebras that possess a multiplicative unit, the spectrum map establishes a contravariant functor (a structure-preserving map between categories) from the category of commutative C*-algebras with unit and unit-preserving continuous *-homomorphisms, to the category of compact Hausdorff spaces and continuous maps. This particular functor is not merely a mapping; it forms one half of a contravariant equivalence between these two categories. Its adjoint is the functor that, somewhat circularly but elegantly, assigns to each compact Hausdorff space $X$ the C*-algebra $C(X)$ (the algebra of all continuous complex-valued functions on $X$ ). This equivalence is a statement of deep structural identity: it implies that the study of commutative C*-algebras is, in a very real sense, equivalent to the study of compact Hausdorff spaces. A direct consequence, and a beautiful illustration of this equivalence, is that for two compact Hausdorff spaces $X$ and $Y$ , the algebra $C(X)$ is isomorphic to $C(Y)$ (as a C*-algebra) if and only if $X$ is homeomorphic to $Y$ . This provides a powerful algebraic method for distinguishing topological spaces.

It is important, for the sake of completeness, to distinguish this commutative Gelfand–Naimark theorem from the more general, "full" Gelfand–Naimark theorem. The latter is a monumental result that applies to arbitrary (abstract) noncommutative C*-algebras $A$ . While not quite analogous to the direct representation as continuous functions, it nonetheless provides a concrete representation of $A$ as an algebra of bounded operators on some Hilbert space. This extension is crucial for areas like quantum mechanics, where non-commutativity is the norm.

Applications

Finally, we arrive at the practical utility, for those who demand such things. One of the most significant applications arising directly from the Gelfand representation is the existence of a continuous functional calculus for normal elements within a C*-algebra $A$ . An element $x$ in a C*-algebra is deemed "normal" if and only if it commutes with its adjoint $x^{*}$ (i.e., $xx^{*} = x^{*}x$ ), or, equivalently, if and only if it generates a commutative C*-algebra, which we denote $C^{*}(x)$ . The fact that it generates a commutative subalgebra is the crucial link, as it allows the Gelfand isomorphism to be brought to bear.

By applying the Gelfand isomorphism to this commutative subalgebra $C^{*}(x)$ , we find that $C^{*}(x)$ is *-isomorphic to an algebra of continuous functions on a locally compact space (specifically, the spectrum of $x$ ). This observation leads almost immediately to a powerful and widely used theorem:

Theorem. Let $A$ be a C*-algebra with an identity element and let $x$ be a normal element of $A$ . Then there exists a unique *-morphism $f \rightarrow f(x)$ from the algebra of continuous functions on the spectrum $\sigma(x)$ into $A$ such that:

It maps the constant function $1$ to the multiplicative identity of $A$ .
It maps the identity function on the spectrum (i.e., the function $g(\lambda) = \lambda$ ) to the element $x$ .

This theorem is profoundly useful. It means that for any normal element $x$ in a C*-algebra, we can meaningfully "plug in" continuous functions. For instance, if $x$ is a normal operator on a Hilbert space, this functional calculus allows us to define functions of operators, such as $e^x$ , $\sqrt{x}$ , or $\cos(x)$ , in a rigorous and consistent manner. This capability is indispensable in numerous fields, from quantum mechanics (where observables are often represented by normal operators) to the study of differential equations and various areas of mathematical physics. It provides a bridge between the abstract world of operators and the more intuitive world of functions, allowing powerful analytical tools to be applied where they might not otherwise seem applicable.