Gelfand Theory and Commutative C Star Algebras

An associative complex algebra is a complex vector space $A$ (Vector space over a field) equipped with a multiplication $A\times A\to A$, $(a,b)\mapsto ab$, which is complex-bilinear and associative: $(\lambda a+\mu b)c=\lambda ac+\mu bc$, $a(\lambda b+\mu c)=\lambda ab+\mu ac$, and $(ab)c=a(bc)$ for all $a,b,c\in A$ and $\lambda ,\mu \in \mathbb{C}$. No multiplicative identity is assumed, and no scalar-algebra or real-algebra convention is imported here; all algebras in this page are complex and the only structure used below is the one just displayed.

A character on $A$ is a map $\chi :A\to \mathbb{C}$ which is nonzero, complex-linear (Linear map between vector spaces over the same field) and multiplicative:

$\chi (\lambda a+\mu b)=\lambda \chi (a)+\mu \chi (b),\chi (ab)=\chi (a)\chi (b)(a,b\in A,\lambda ,\mu \in \mathbb{C})\mathrm{.}$

Two things are deliberately not part of the definition:

• continuity is not assumed; for a unital Banach algebra it is a theorem below, and for a commutative Banach algebra it then follows for free;
• preservation of a unit is not assumed either. If $A$ happens to have an identity $1$, a character is not required to satisfy $\chi (1)=1$ by definition; for a unital Banach algebra this too is proved later on this page.

The character space of $A$ is

$\mathrm{\Delta }(A):=\{\chi :\chi \text{ is a character on }A\},$

a set by Separation, since every character is a subset of $A\times \mathbb{C}$ and $A\times \mathbb{C}$ is a set. Initially $\mathrm{\Delta }(A)$ carries the topology of pointwise evaluation: the coarsest topology for which all the evaluation maps $e_a:\mathrm{\Delta }(A)\to \mathbb{C}$, $e_a(\chi ):=\chi (a)$, are continuous. Equivalently, a basic neighbourhood of ${\chi }_0$ is $\{\chi :\mathrm{\mid }\chi (a_i)-{\chi }_0(a_i)\mathrm{\mid }<\epsilon ,i\le k\}$ for finitely many $a_i\in A$ and $\epsilon >0$. Once characters are known to be bounded linear functionals they are points of the dual $A^{\ast }$, and this topology is exactly the subspace topology induced by the weak-star topology $\sigma (A^{\ast },A)$ (as proved later on this page); no duality theory is used before that point. For a nonzero commutative unital Banach algebra, under the Axiom of Choice, $\mathrm{\Delta }(A)$ is also called the maximal ideal space: only in that setting does the later maximal-ideal correspondence identify its points with all maximal ideals.

• Why "nonzero" is part of the definition. The zero map is linear and multiplicative and would otherwise be a character of every algebra; excluding it is what makes characters the algebraic counterparts of points, and it is used already in the first unitality computation.
• The empty character space is allowed here. For an algebra with no characters at all $\mathrm{\Delta }(A)=\mathrm{\varnothing }$ is a perfectly good value of the definition; compact Hausdorffness and nonemptiness of $\mathrm{\Delta }(A)$ are theorems requiring a nonzero commutative unital Banach algebra.
• A character need not exist. For a general associative complex algebra nothing in this definition produces a character, and the existence statements below spend the Axiom of Choice precisely there.

Let $A$ be a nonzero unital complex Banach algebra (Unital Banach algebra) and let $\chi$ be a character on $A$ (Character and maximal ideal space). Then:

1. $\chi$ is unital: $\chi (1)=1$;
2. $\chi (a)\in {\sigma }_A(a)$ for every $a\in A$ (Spectrum and resolvent set in a Banach algebra);
3. $\mathrm{\mid }\chi (a)\mathrm{\mid }\le \mathrm{\parallel }a\mathrm{\parallel }$ for every $a\in A$; consequently $\chi$ is bounded linear with $\mathrm{\parallel }\chi \mathrm{\parallel }=1$, and in particular $\chi$ is continuous.

No Hahn–Banach theorem, no spectral-radius formula and no existence of characters is used: the argument runs on the Neumann series alone and is a theorem of ZF.

• The unit hypothesis streamlines the proof, but continuity survives without it. For a nonzero unital Banach algebra the computation $\chi (1{)}^2=\chi (1)$ forces unitality. A character on a nonunital Banach algebra $B$ extends to the algebraic sum-norm unitization $\mathbb{C}\oplus B$, with product $(\lambda ,a)(\mu ,b)=(\lambda \mu ,\lambda b+\mu a+ab)$, by ${\chi }^{+}(\lambda ,a)=\lambda +\chi (a)$. This is a unital character on a unital Banach algebra, so the theorem applied to ${\chi }^{+}$ shows that the original character is continuous as well.
• Choice-free. Steps 1.1–2.2 use only the algebra axioms, the definition of the spectrum and the Neumann series; no selection from nonempty sets and no separation theorem occurs.
• Where the bound is used. Part 3 is what puts $\mathrm{\Delta }(A)$ inside the dual unit ball and identifies the pointwise-evaluation topology with the weak-star subspace topology; this is the standard automatic-continuity statement for characters.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $A$ be a unital complex Banach algebra (Unital Banach algebra) and let $I\subseteq A$ be a proper two-sided ideal which is closed in the norm of $A$. Then the quotient $A\mathrm{/}I$, with the quotient norm and coset multiplication, is a nonzero unital complex Banach algebra with unit $1+I$ of norm one, and the quotient map $A\to A\mathrm{/}I$ is a unital algebra homomorphism of norm one.

The hypothesis of Countable Choice is inherited from the completeness of the quotient (A quotient of a Banach space by a closed subspace is Banach) and is spent nowhere else; the algebraic verification is a theorem of ZF.

• Why the ideal must be closed. Completeness of the quotient is exactly what fails for a non-closed ideal; the argument above uses closedness only through [L3].
• Countable Choice is genuinely used. The completeness of the quotient is inherited from the published quotient theorem, which assumes ${\mathrm{A}\mathrm{C}}_{\omega }$; no other step selects from infinitely many nonempty sets, and the two near-minimizing representatives in step 2.2 are chosen for a single pair $(a,b)$ at each fixed $\epsilon$.
• Properness is used twice. It gives $1\mathrm{\notin }I$ (nonzero quotient) and the distance bound $\mathrm{\parallel }1+I\mathrm{\parallel }\ge 1$ through the Neumann series.

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a nonzero commutative unital complex Banach algebra (Unital Banach algebra) and let $I\subseteq A$ be a proper ideal. Then:

1. $I$ is contained in a closed maximal ideal;
2. the maximal ideals of $A$ (Prime ideals and maximal ideals in a commutative ring) are exactly the kernels of the characters of $A$ (Character and maximal ideal space), and the map $\chi \mapsto \ker \chi$ is a bijection from $\mathrm{\Delta }(A)$ onto the set of maximal ideals.

This is an implementation in ZFC: every proper ideal is extended by Zorn's lemma, so the argument is not an equivalence between the existence of maximal ideals and a weaker choice principle.

• Where the Axiom of Choice is spent. Directly through Zorn's lemma in [step 2.1], applied to the poset built in [step 1.3]; through the Countable Choice used for the completeness of the quotient in [step 3.2], which [L7] derives from AC; and through Gelfand–Mazur in [step 4.1], whose nonemptiness-of-spectrum input spends AC. The remaining computations of [step 1.1], [step 1.2], [step 1.4] and [step 1.5] involve no further selection.
• Maximal ideals are automatically closed. This is what makes the maximal ideal space a topological object: by [2.1] the word "closed" in claim 1 is redundant, but it is proved, not assumed.
• No unit is assumed on the quotient. The quotient unit is $1+M$, whose norm is one by Closed ideal quotient is a Banach algebra; the nonzero hypothesis on $A$ is used only to know that the zero ideal is proper.

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a nonzero commutative unital complex Banach algebra (Unital Banach algebra), let $a\in A$, and let ${\sigma }_A(a)$ be the spectrum of $a$ in $A$ (Spectrum and resolvent set in a Banach algebra), with spectral radius $r(a)$ (Spectral radius). Then

${\sigma }_A(a)=\{\chi (a):\chi \in \mathrm{\Delta }(A)\},$

and consequently every character $\chi \in \mathrm{\Delta }(A)$ satisfies $\mathrm{\mid }\chi (a)\mathrm{\mid }\le r(a)\le \mathrm{\parallel }a\mathrm{\parallel }$ for every $a\in A$.

The spectrum is taken in the ambient algebra $A$; the statement is not a claim about the spectrum computed in a subalgebra, and no injectivity or surjectivity of the Gelfand transform is asserted.

• AC is spent once, in the maximal-ideal extension. Both inclusions are otherwise algebraic: the forward inclusion only tests the character on a coset representative, and the reverse inclusion only extends an ideal.

• Consequences for the Gelfand transform. Since $\widehat{a}(\chi )=\chi (a)$, the equality of the statement says that the range of $\widehat{a}$ is exactly ${\sigma }_A(a)$, which is how the norm formula $\mathrm{\parallel }\widehat{a}{\mathrm{\parallel }}_{\mathrm{\infty }}=r(a)$ is proved in Gelfand transform is a contractive unital homomorphism.

• No isometry claim. The inequality chain $\mathrm{\mid }\chi (a)\mathrm{\mid }\le r(a)\le \mathrm{\parallel }a\mathrm{\parallel }$ is all that the spectrum identity yields; for a general commutative Banach algebra the first inequality can be strict, as cex-gelfand-transform-of-a-banach-algebra-need-not-be-isometric records.

• Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a nonzero commutative unital complex Banach algebra (Unital Banach algebra) and let $\mathrm{\Delta }(A)$ be its character space (Character and maximal ideal space). Then:

1. each character of $A$ is a bounded linear functional of norm one, so that $\mathrm{\Delta }(A)$ is a subset of the closed dual unit ball $B_{A^{\ast }}=\{f\in A^{\ast }:\mathrm{\parallel }f\mathrm{\parallel }\le 1\}$;
2. the pointwise-evaluation topology of $\mathrm{\Delta }(A)$ is the subspace topology induced by the weak-star topology $\sigma (A^{\ast },A)$;
3. $\mathrm{\Delta }(A)$ is a weak-star closed subset of $B_{A^{\ast }}$;
4. $\mathrm{\Delta }(A)$ is nonempty, compact and Hausdorff, hence a compact Hausdorff space in the weak-star topology, and it is a nonempty compact Hausdorff subset of the dual unit ball in that topology.

The Axiom of Choice enters exactly through the ultrafilter lemma (for Banach–Alaoglu) and through the existence of maximal ideals (for nonemptiness); no further selection is made.

• Compactness is a weak-star statement. Banach–Alaoglu is applied to $B_{A^{\ast }}$ with no completeness hypothesis on $A$; the only use of completeness is through the continuity and norm bound of characters, and the only use of commutativity is through the maximal ideal theorem.
• Nonemptiness is not automatic. For a commutative unital Banach algebra over $\mathbb{C}$ it is a consequence of Zorn; the proof does not construct a character explicitly, and the case of the zero algebra is excluded by the hypothesis that $A$ is nonzero.
• The two topologies agree on $\mathrm{\Delta }(A)$ only because characters are bounded. Before the automatic-continuity theorem the pointwise-evaluation topology of $\mathrm{\Delta }(A)$ is not a weak-star subspace topology, since $\mathrm{\Delta }(A)$ is not a subset of the dual.

Let $A$ be a commutative unital complex algebra and let $\mathrm{\Delta }(A)$ be its character space with the pointwise-evaluation topology (Character and maximal ideal space). For $a\in A$ define its Gelfand transform $\widehat{a}:\mathrm{\Delta }(A)\to \mathbb{C}$ by

$\widehat{a}(\chi ):=\chi (a)(\chi \in \mathrm{\Delta }(A)),$

and define the Gelfand transform of $A$ as the map

${\mathrm{\Gamma }}_A:A⟶{\mathbb{C}}^{\mathrm{\Delta }(A)},{\mathrm{\Gamma }}_A(a):=\widehat{a}\mathrm{.}$

Each $\widehat{a}$ is continuous for the pointwise-evaluation topology, because that topology is by definition the coarsest one making every evaluation $\chi \mapsto \chi (b)$ continuous and $\widehat{a}$ is the evaluation at $a$; the codomain is written ${\mathbb{C}}^{\mathrm{\Delta }(A)}$ with the product topology, and the image of ${\mathrm{\Gamma }}_A$ therefore lies in the algebra $C(\mathrm{\Delta }(A))$ of continuous functions. No supremum norm or compactness is asserted for a general $A$. If $A$ is in addition a nonzero unital commutative Banach algebra and the Axiom of Choice is assumed (The Axiom of Choice), then Maximal ideal space is compact Hausdorff makes $\mathrm{\Delta }(A)$ compact and $C(\mathrm{\Delta }(A))$ carries its supremum norm.

The formula also makes sense if $\mathrm{\Delta }(A)$ is empty: every transform is the unique function on the empty set. It sends $0$ to the zero function. For any character choose $b$ with $\chi (b)\mathrm{\ne }0$; the equality $\chi (b)=\chi (1)\chi (b)$ gives $\chi (1)=1$. Thus the unit maps to the constant-one function (also well defined on an empty character space).

Two qualifications are part of the definition:

• the notation is introduced for unital commutative algebras here; the nonunital version with target $C_0(\mathrm{\Delta }(A))$ for commutative C*-algebras under AC is a theorem proved later (Nonunital commutative Gelfand Naimark) and is not smuggled into the definition;
• no injectivity, surjectivity, isometry or *-preservation is claimed at this point. Those properties are theorems, valid under progressively stronger hypotheses, and the map ${\mathrm{\Gamma }}_A$ is defined for every commutative unital complex algebra.

• Notation. We write $\widehat{a}$ for ${\mathrm{\Gamma }}_A(a)$ and drop the subscript $\mathrm{\Gamma }={\mathrm{\Gamma }}_A$ when the algebra is clear; the algebra, not the element, is what $\mathrm{\Gamma }$ encodes.
• Values are character values. If $A$ is a nonzero commutative unital complex Banach algebra and AC is assumed, the identity ${\sigma }_A(a)=\{\chi (a):\chi \in \mathrm{\Delta }(A)\}$ of Spectrum as character values reads $\mathrm{ran}(\widehat{a})={\sigma }_A(a)$, which is the form in which the spectrum will be computed from the transform.

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a nonzero commutative unital complex Banach algebra (Unital Banach algebra) with Gelfand transform $\mathrm{\Gamma }={\mathrm{\Gamma }}_A:A\to C(\mathrm{\Delta }(A))$ (Gelfand transform). Then:

1. $\mathrm{\Gamma }$ is a unital complex-algebra homomorphism: $\mathrm{\Gamma }(\lambda a+\mu b)=\lambda \mathrm{\Gamma }(a)+\mu \mathrm{\Gamma }(b)$, $\mathrm{\Gamma }(ab)=\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)$ and $\mathrm{\Gamma }(1)=\mathbf{1}$;
2. $\mathrm{\parallel }\mathrm{\Gamma }(a){\mathrm{\parallel }}_{\mathrm{\infty }}=r(a)$ for every $a\in A$ (Spectral radius), and consequently $\mathrm{\Gamma }$ is contractive: $\mathrm{\parallel }\mathrm{\Gamma }(a){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }a\mathrm{\parallel }$.

No injectivity, surjectivity or *-preservation is claimed for a general commutative unital Banach algebra.

• The sup norm is finite. Boundedness of $\widehat{a}$ is not assumed: it follows from the norm bound $\mathrm{\mid }\chi (a)\mathrm{\mid }\le \mathrm{\parallel }a\mathrm{\parallel }$ of [L1], so the supremum in $\mathrm{\parallel }\mathrm{\Gamma }(a){\mathrm{\parallel }}_{\mathrm{\infty }}$ is taken over a bounded set of values.
• The formula $\mathrm{\parallel }\widehat{a}{\mathrm{\parallel }}_{\mathrm{\infty }}=r(a)$ is the exact quantitative content of the theorem; the inequality $r(a)\le \mathrm{\parallel }a\mathrm{\parallel }$ is the contractivity, and for a general Banach algebra it may be strict.

Let $A$ be a commutative unital complex algebra (Character and maximal ideal space for the algebra convention and Prime ideals and maximal ideals in a commutative ring for ideals). The Jacobson radical of $A$ is

$\mathrm{rad}(A):=\bigcap \{M:M\text{ is a maximal ideal of }A\},$

the intersection of all maximal ideals of $A$; when $A$ has no maximal ideals the intersection is over the empty family and the radical is $A$ by the convention that an empty intersection is the whole ring. The algebra $A$ is called semisimple when

$\mathrm{rad}(A)=\{0\}\mathrm{.}$

The radical is an ideal: it is the intersection of a family of ideals, hence closed under addition and under multiplication by arbitrary elements of $A$. The definition is stated for commutative unital complex algebras only, which is the class for which the radical is used in this page; it is not the general noncommutative Jacobson radical, and no noncommutative radical characterization is invoked anywhere in this library's Gelfand theory.

• Two equivalent readings for Banach algebras under Choice. Assuming the Axiom of Choice (The Axiom of Choice), for a commutative unital Banach algebra the intersection of all maximal ideals is the same as the intersection of the kernels of all characters, because the maximal ideals are exactly the character kernels (Maximal ideals and characters of a commutative Banach algebra); this equality is used in Kernel of the Gelfand transform is the radical, not assumed here.
• Semisimplicity is an algebraic condition. It says that the maximal ideals separate points of $A$ in the weak sense of having trivial intersection; it does not by itself say anything about the norm, and it is compatible with $\mathrm{\Gamma }$ failing to be isometric.

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a nonzero commutative unital complex Banach algebra (Unital Banach algebra) with Gelfand transform $\mathrm{\Gamma }={\mathrm{\Gamma }}_A:A\to C(\mathrm{\Delta }(A))$ (Gelfand transform) and Jacobson radical $\mathrm{rad}(A)$ (Jacobson radical and semisimple commutative Banach algebra). Then

$\ker \mathrm{\Gamma }={\bigcap }_{\chi \in \mathrm{\Delta }(A)}\ker \chi =\mathrm{rad}(A)\mathrm{.}$

Consequently $\mathrm{\Gamma }$ is injective if and only if $A$ is semisimple. No claim about $\mathrm{\Gamma }$ being isometric or surjective is made; injectivity of $\mathrm{\Gamma }$ is a statement about the radical only.

• The two intersections in the statement are the same set for two different reasons. The first is the definition of $\mathrm{\Gamma }$ evaluated at zero, the second is the maximal ideal theorem; the theorem is the equality of the two descriptions.
• Semisimplicity still does not give isometry. By Gelfand transform is a contractive unital homomorphism one always has $\mathrm{\parallel }\widehat{a}{\mathrm{\parallel }}_{\mathrm{\infty }}=r(a)\le \mathrm{\parallel }a\mathrm{\parallel }$, and semisimplicity upgrades injectivity of $\mathrm{\Gamma }$, not the norm equality $\mathrm{\parallel }\widehat{a}{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }a\mathrm{\parallel }$.

A possibly nonunital complex Banach algebra is a complex vector space $A$ equipped with an associative complex-bilinear multiplication $A\times A\to A$ (A bounded bilinear map between normed spaces for the bilinearity convention) and a norm under which $A$ is a Banach space (Banach space, Real and complex scalar conventions for normed spaces), such that the norm is submultiplicative:

$\mathrm{\parallel }ab\mathrm{\parallel }\le \mathrm{\parallel }a\mathrm{\parallel }\mathrm{\parallel }b\mathrm{\parallel }(a,b\in A)\mathrm{.}$

No multiplicative identity is assumed, and no real-algebra or unital convention is imported from elsewhere.

A complex C*-algebra is a possibly nonunital complex Banach algebra $A$ equipped with an involution $A\to A$, $a\mapsto a^{\ast }$, which is conjugate-linear (Real and imaginary parts, complex conjugation, and modulus) and satisfies