nLab
Banach algebra

Definitions

An associative unital Banach algebra is monoid object in the category of Banach spaces. However, Banach algebras are not usually assumed to be unital, making them semigroup objects (or even magma objects if not assumed to be associative).

This means a Banach space $A$ equipped with a multiplication map

$$m:A\otimes A\to A,$$m: A \otimes A \to A ,

where $\otimes$ is the standard tensor product for the symmetric monoidal closed category of Banach spaces, which again is usually taken to be associative.

Since the norm of $a\otimes b\in A\otimes A$ is $\parallel a\parallel \cdot \parallel b\parallel$ where $\parallel a\parallel$ is the norm in $A$, and since morphisms in the category of Banach spaces are taken to be bounded linear maps of norm $\le 1$, we may equivalently define a Banach algebra to be equipped with a bilinear map (usually associative) $A\times A\to A$ such that

$$\parallel a\cdot b\parallel \le \parallel a\parallel \cdot \parallel b\parallel$$\|a \cdot b\| \leq \|a\| \cdot \|b\|

where $a\cdot b=m(a,b)$.

Of course, in the non-unital case, one can always formally adjoin a unit $e$, forming the Banach algebra $A\oplus ⟨e⟩$ where $\parallel e\parallel =1$.

Examples

• A standard example is $L^1(ℝ,\mu )$, where $\mu$ is Lebesgue measure, and where the multiplication is taken to be convolution. This lacks a unit for the multiplication, since there is no $L^1$ function $e(x)$ that represents the Dirac functional

  $$f\mapsto f(0)=\int e(x)f(x)d\mu$$f \mapsto f(0) = \int e(x)f(x) d\mu

  on continuous functions $f:X\to ℂ$. One can generalize this example in straightforward fashion, replacing $ℝ$ by any locally compact Hausdorff topological group $G$, and $\mu$ by a Haar measure? on $G$; the algebra is unital if and only if $G$ is compact.

• For any measure space $(X,\mu )$, $L^{\mathrm{\infty }}(X,\mu )$ is a unital Banach algebra with respect to pointwise multiplication.

• If $A$ is a Banach space, the internal hom $\mathrm{hom}(A,A)$ is a unital Banach algebra.

• Any $C^{*}$-algebra is a Banach algebra.

• The normed division algebras are (possibly nonassiociative) Banach algebras that are also division algebra?s.

• The only Banach algebra (over $ℂ$) which is a field is $ℂ$ itself, by the Gel’fand–Mazur theorem.