The Gelfand Representation for C*-Algebras

The Gelfand Representation is an important topic in functional analysis. This text collects and orders important properties, with a focus on the basics of functional calculus for C*-algebras. Much of this discussion also applies to Banach algebras.

The Character Space

Let $A$ be a C*-algebra, and consider the set of characters

\[\Omega(A) = \{ \chi : A \to \mathbb{C} \; | \;\chi\; \text{is a non-zero homomorphism} \}.\]

This is called the character space, or spectrum, of $A$ (sometimes written $\text{Spec}\: A$). Given an element $a \in A$, we define the Gelfand transform of $a$ as the evaluation map

\[\hat{a} : \Omega(A) \to \mathbb{C}, \;\;\; \chi \longmapsto \chi(a).\]

We endow the character space $\Omega(A)$ with the weakest topology that ensures each element $\hat{a}$ is continuous. Take an open set $U \subset \mathbb{C}$, the preimage is then

\[\hat{a}^{-1}(U) = \{ \chi \in \Omega(A) : \chi(a) \in U \}.\]

We take all such preimages (for all elements $a \in A$ and all open sets $U$ in $\mathbb{C}$) as a subbase for the topology. The topology we use is the smallest topology containing the subbase as open sets. This is exactly the weak* topology on $\Omega(A)$.

Aside: Weak vs Weak* Topology

Given an algebra $A$, its character space $\Omega(A)$ is a subset of the topological dual space $A^*$ (all bounded linear maps from $A$ to $\mathbb{C}$). The weak* topology on $A^*$ is the smallest topology that keeps $\hat{a} : f \mapsto f(a)$ continuous for every $a \in A$.

Each $\hat{a}$ is an element of the double dual $A^{**}$, that is, a bounded linear map from $A^*$ to $\mathbb{C}$. So the weak* topology requires a subset of $A^{**}$ to be continuous. The weak topology on the other hand requires every single element of $A^{**}$ to be continuous. This is a stronger requirements than for the weak* topology. A subbasis for the weak topology is given by sets of the form $$ \Phi^{-1}(U) = \{ \chi \in \Omega(A) : \Phi(\chi) \in U \}. $$ Where $\Phi$ is an element of $A^{**}$ and $U \subset \mathbb{C}$.

The Spectrum

Let $A$ be a commutative C*-algebra, then the character space is closely related to the spectrum of each element.

Unital Algebra

For $a \in A$, $\chi \in \Omega(A)$, let $\lambda = \chi(a)$. Assume $A$ is unital, then

\[\chi(a - \lambda 1) = \chi(a) - \lambda = 0\]

Now, if $a - \lambda 1$ is invertible, let $b$ be the inverse. Then

\[1 = \chi((a - \lambda 1)b) = \chi(a - \lambda 1)\chi(b) = 0,\]

a contradiction, hence $\lambda = \chi(a) \in \sigma(a)$.

For the other direction, let $\lambda \in \sigma(a)$ and consider the set $I = (a - \lambda)A$. We have $1 \not \in I$, and for $b \in A$, $bI = Ib = I$ by commutativity of $A$. That is, $I$ is a two-sided proper ideal of $A$.

Every proper ideal is contained in a proper maximal ideal $M$. Then the quotient $A/M$ is a simple commutative C*-algebra. That is, it is isomorphic to $\mathbb{C}$.

Hence $A\cong M \oplus \mathbb{C}$. We can thus define a character $\tau : A \to \mathbb{C}$ by $\tau(m, \lambda) = \lambda$ so that $M = \text{ker}(\tau)$. As $a - \lambda \in M$, $\tau(a - \lambda) = 0$, meaning $\tau(a) = \lambda$.

Hence we have that for $A$ commutative and unital

\[\sigma(a) = \{ \chi(a) \; : \; \chi \in \Omega(A)\}.\]

Non-Unital Algebra

When $A$ is non-unital, we define the spectrum of an element $a \in A$ by the spectrum in the unitisation $\tilde{A} = A \oplus \mathbb{C}$ of $A$. Then the character space on $\tilde{A}$ is

\[\Omega(\tilde{A}) = \{ \tilde{\chi} \; : \; \chi \in \Omega(A) \} \cup \{\tau\}.\]

Where $\tilde{\chi}(a, \lambda) = \chi(a) + \lambda$ and $\tau(a, \lambda) = \lambda$. Hence we have

\[\sigma(a) = \{ \chi(a) \; : \; \chi \in \Omega(A)\} \cup \{0\}\]

The Character Space is Locally Compact

Let $A$ be Given a character $\chi \in \Omega(A)$, it is an exercise to show that $\lVert \chi \rVert \leq 1$. Importantly, the character space $\Omega(A)$ is contained within the closed unit ball of $A^*$. By Banach-Alaoglu, the closed unit ball in the weak* topology is compact. Thus, every element of the character space has a compact neighbourhood, and so $\Omega(A)$ is locally compact.

If we assume $A$ is unital, then let $\chi \in \Omega(A)$, we have $\chi(1a) = \chi(1)\chi(a)$, so $\chi(1) = 1$ for every non-zero character. Thus, any net of characters $(\chi_\lambda)_{\lambda\in\Lambda}$ in $\Omega(A)$ must converge to a functional $\chi \in A^*$ with $\chi(1) = 1$.

To show this is a character, let $a, b \in A$, then

\[\chi(ab) = \lim_\lambda \chi_\lambda(ab)= \lim_\lambda \chi_\lambda(a)\chi_\lambda(b) = \lim_\lambda \chi_\lambda(a)\lim_\lambda\chi_\lambda(b) = \chi(a)\chi(b),\]

as weak* convergence means $\chi(a) = lim_\lambda \chi_\lambda(a)$ for all $a \in A$. Thus if $A$ is a commutative unital C*-algebra the spectrum $\Omega(A)$ is weak*-closed and hence compact

Thus the space $C(\Omega(A))$, for $A$ unital, is a C*-algebra.

For $A$ non-unital we instead write $C_0(\Omega(A))$, that is, the continous functions on $\Omega(A)$ which vanish as they go to infinity. This is also a C*-algebra.

Commutative Gelfand-Naimark Theorem

Unital

Let $A$ be a commutative unital C*-algebra. We define a norm on $C(\Omega(A))$ by

\[|| \psi ||_\infty = \sup_{\chi\in\Omega(A)} ||\psi(\chi)||\]

Let $\phi : A \to C(\Omega(A))$ be the map to the Gelfand representation of each element of $A$.

\[\phi(a) = \hat{a}.\]

Then

\[|| \phi(a) ||_\infty = \sup_{\chi\in\Omega(A)} ||\phi(a)\chi|| = \sup_{\chi\in\Omega(A)} ||\chi(a)|| = \sup_{\lambda\in\sigma(a)} |\lambda| = r(a).\]

We call $r(a)$ the spectral radius of $a$. For C*-algebras, we have $r(a) = ||a||$, thus $||\phi(a)|| = r(a) = ||a||$, so $\phi$ is an isometry.

If we take $a, b \in A$ and $\chi \in \Omega(A)$, we have

\[(\phi(a)\phi(b))\chi = (\hat{a}\hat{b})\chi = \chi(a)\chi(b) = \chi(ab) = \hat{ab}\chi = \phi(ab)\chi\]

and

\[\phi(a + b)\chi = \chi(a) + \chi(b) = \chi(a + b) = \phi(a + b)\chi\]

finally

\[\phi(a^*)\chi = \chi(a^*) = \overline{\chi(a)} = \overline{\phi(a)\chi}\]

so that $\phi$ is a *-homomorphism.

Let $X = \phi(A)$, then $X$ is a unital closed sub-*-algebra of $C(\Omega(A))$. Let $\chi \neq \rho \in \Omega(A)$, then there is an $a \in A$ with $\chi(a) \neq \rho(a)$, thus $\phi(a)\chi \neq \phi(a)\rho$. So $B$ separates points in $\Omega(A)$. Thus we can use the Stone-Weierstrass theorem to conclude that $B = C(\Omega(A))$.

Thus $\phi$ is an isometric *-isomorphism, so we can write $A \cong C(\Omega(A))$.

Non-Unital

Let $A$ be a commutative C*-algebra, and consider the unitisation $\tilde{A}$. For a character $\chi \in \Omega(A)$ we define a character on the unitisation

\[\tilde{\chi}(a + \lambda 1) = \chi(a) + \lambda \in \mathbb{C}.\]

such that