nLab
Banach algebra

Definitions

An associative unital Banach algebra is monoid object in the closed monoidal category of Banach spaces (with short linear operators as morphisms, and the usual internal hom, or equivalently the projective tensor product). However, Banach algebras are not usually assumed to be unital, making them semigroup objects (or even magma objects if not assumed to be associative).

Explicitly, this means a Banach space $A$ equipped with a bilinear multiplication map

$$m:A\times A\to A,$$m\colon A \times A \to A ,

which again is usually taken to be associative (and may even be unital), such that

$$\parallel a\cdot b\parallel \le \parallel a\parallel \cdot \parallel b\parallel ,$${\|a \cdot b\|} \leq {\|a\|} \cdot {\|b\|} ,

where $a\cdot b$ (or just $ab$) means $m(a,b)$.

Of course, in the non-unital case, one can always formally adjoin a unit $e$ with $\parallel e\parallel =1$, forming the Banach algebra $A\oplus ⟨e⟩$ (using the $l^1$-direct sum).

The explicit description in terms of $m$ is of course earlier; but the abstract description as an internal monoid makes clear the correct definition of Banach coalgebra: a comonoid in the same monoidal category.

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 (in fact a $C^{*}$-algebra) with respect to pointwise multiplication.

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

• Any C-star algebra is in particular a Banach algebra.

• The normed division algebras are (possibly nonassiociative) Banach division algebras over $ℝ$.

• The only Banach division algebra over $ℂ$ is $ℂ$ itself, by the Gel’fand–Mazur theorem.

References

• Zbigniew Semadeni, Banach spaces of continuous functions, vol. I, gBooks
• N. Landsman, Mathematical topics between classical and quantum mechanics, Springer
• Walter Rudin, Functional analysis
• Richard V. Kadison, John R. Ringrose, Fundamentals of the theory of operator algebras

category: analysis