Unital Banach algebra

A unital complex Banach algebra is a nonzero complex vector space $A$ equipped with

• an associative bilinear multiplication $A\times A\to A$, $(a,b)\mapsto ab$, and
• a norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ under which $A$ is a Banach space (Banach space),

such that

• the norm is submultiplicative: $\mathrm{\parallel }ab\mathrm{\parallel }\le \mathrm{\parallel }a\mathrm{\parallel }\mathrm{\parallel }b\mathrm{\parallel }$ for all $a,b\in A$, and
• there is a unit $1\in A$ with $1a=a1=a$ for every $a\in A$, normalized by $\mathrm{\parallel }1\mathrm{\parallel }=1$.

The phrase nonzero is part of the definition: the identity is required to exist, and the zero algebra $\{0\}$ has no element satisfying $1\mathrm{\ne }0$. In every unital Banach algebra the unit is unique, because if $1^{\mathrm{\prime }}$ is a second element acting as an identity then $1^{\mathrm{\prime }}=1^{\mathrm{\prime }}1=1$; from now on $1$ denotes that element. The norm condition $\mathrm{\parallel }1\mathrm{\parallel }=1$ is a normalization rather than a consequence of submultiplicativity, which would give only $\mathrm{\parallel }1\mathrm{\parallel }\ge 1$ for a nonzero unit; the two conventions $\mathrm{\parallel }1\mathrm{\parallel }=1$ and $\mathrm{\parallel }1\mathrm{\parallel }\le 1$ agree on every nonzero unital algebra, since submultiplicativity turns the latter into an equality.

• The scalar field is complex and fixed. Every spectrum, resolvent and holomorphic-calculus statement on this page is about complex unital Banach algebras. Real Banach algebras are not silently complexified: the one place where a real structure enters, the spectrum of a real operator, is defined through a specified complexification in Complexification and spectrum of a real operator.

• Multiplication is bilinear and associative, and nothing more. No commutativity, involution, or approximate unit is assumed. The algebra $\mathcal{B}(X)$ of bounded operators on a nonzero complex Banach space (ex-bounded-operators-form-a-noncommutative-banach-algebra) is the motivating noncommutative example, and $C(K,\mathbb{C})$ for compact Hausdorff $K$ (ex-continuous-functions-form-a-commutative-banach-algebra) the motivating commutative one.

• Completeness is with respect to the submultiplicative norm. A complete normed algebra whose norm is merely equivalent to a submultiplicative one is not thereby a unital Banach algebra in this sense; rescaling a norm to $\lambda \mathrm{\parallel }\cdot \mathrm{\parallel }$ with $\lambda >1$ preserves completeness and submultiplicativity but destroys the normalization $\mathrm{\parallel }1\mathrm{\parallel }=1$.

• The unit is not a separate structure. It is determined by the multiplication, so an algebra homomorphism between unital Banach algebras that preserves multiplication and the unit is exactly a multiplicative linear map sending $1$ to $1$; this is the convention used for characters on the following page of this track.

• 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.