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A Banach algebra is an associative algebra over $\mathbb{R}$ or $\mathbb{C}$ whose norm makes it a Banach space and satisfies submultiplicativity:An identity is an additional assumption; when present, its normalization $\Vert 1\Vert =1$ should be stated if needed. Submultiplicativity makes multiplication continuous, and completeness makes norm-controlled infinite algebraic operations possible.^[ref]

Examples include bounded linear operators on a Banach space with composition and the operator norm, and continuous complex-valued functions on a compact Hausdorff space with pointwise multiplication and the supremum norm. The former is generally noncommutative, whereas the latter is commutative. A closed subalgebra of a Banach algebra inherits both the product estimate and completeness.

For $\Vert a\Vert <1$, the powers satisfy ${\sum }_n\Vert a^n\Vert <\infty$, so the Neumann series constructs $(1-a{)}^{-1}$. This simple estimate connects analysis to invertibility and leads to the spectrum of an element and holomorphic functional calculus.

A C-star algebra is a complex Banach algebra with an algebra involution $a\mapsto a^{\ast }$ satisfying the C-star identityThe algebra involution is conjugate-linear, reverses products and obeys $(a^{\ast }{)}^{\ast }=a$. An identity need not be present.^[ref]

Examples include complex matrices with the operator norm and conjugate transpose, bounded linear operators on a Hilbert space with the operator adjoint, and $C_0(X)$ on a Hausdorff space that is a locally compact space with pointwise complex conjugation and the supremum norm. The identity ties the norm to the algebraic product and involution: it implies spectral radius norm equality for normal elements and makes an injective C-star homomorphism an isometry. This rigidity permits continuous functional calculus for normal C-star elements.

Every nonzero complex unital Banach algebra in which every nonzero element is invertible is isomorphic as a complex algebra to $\mathbb{C}$. Indeed, the nonemptiness of the Banach-algebra spectrum gives, for any $a$, some $\lambda \in {\sigma }_A(a)$. The element $a-\lambda 1$ is noninvertible, so the hypothesis forces $a=\lambda 1$. Thus $\lambda \mapsto \lambda 1$ is an algebra isomorphism onto $A$.^[ref]

The isometry conclusion additionally uses $\Vert 1\Vert =1$, since $\Vert \lambda 1\Vert =|\lambda |\Vert 1\Vert$. Without that convention, $\mathbb{C}$ equipped with $\Vert z\Vert =2|z|$ is still a Banach algebra with every nonzero element invertible, but its unital complex algebra isomorphism from the usual $\mathbb{C}$ is not isometric. The complex scalar field is also essential: $\mathbb{R}$ itself gives a real Banach algebra satisfying the invertibility condition.

The unitization of a complex associative algebra $A$ is $A^{+}=A\oplus \mathbb{C}$ with productIts identity is $(0,1)$, and $a\mapsto (a,0)$ identifies $A$ with a two-sided ideal. If $A$ is a Banach algebra, the norm $\Vert (a,\lambda )\Vert =\Vert a\Vert +|\lambda |$ makes $A^{+}$ a unital Banach algebra with the original norm on $A$. The product formula directly proves submultiplicativity.^[ref]

For a noncompact Hausdorff space $X$ that is a locally compact space, adjoining constant functions identifies the unitization of $C_0(X)$ algebraically with $C(X^{+})$, where $X^{+}$ is the one-point compactification. This does not identify the displayed sum norm with the supremum norm: those are equivalent norms, but the C-star unitization uses the latter. Unitization lets definitions such as the spectrum of an element apply to algebras that initially have no identity.

Vahid Shirbisheh authored lectures on C-star algebras connecting Banach algebras, spectral theory and functional calculus. The notes develop the Gelfand transform and the correspondence between commutative C-star algebras and appropriate topological spaces. This explains how an abstract algebra can encode continuous functions and recover geometric information from its multiplicative structure.^[ref]