Banach algebra - Knowlpedia

Let $\mathbb F=\mathbb R$ or $\mathbb C$ . A Banach algebra over $\mathbb F$ is a vector space $A$ over $\mathbb F$ equipped with an associative $\mathbb F$ -bilinear multiplication $A\times A\to A$ and a norm such that $A$ is a Banach space and

\lVert ab\rVert\leq \lVert a\rVert\,\lVert b\rVert

for all $a,b\in A$ .

Continuity and unity

The inequality makes multiplication jointly continuous. This definition does not require an identity. A unital Banach algebra additionally has a multiplicative identity; here its norm is normalized by $\lVert 1\rVert=1$ .

Unitization and spectral notions

Every nonunital complex Banach algebra has a unitization $A^+=A\oplus\mathbb C1$ containing $A$ as a closed ideal. For $a$ in a unital complex Banach algebra, the spectrum is the set of $\lambda\in\mathbb C$ for which $\lambda1-a$ is not invertible. Completeness is essential to the standard theorem that this spectrum is nonempty and compact.

Standard examples

The bounded linear operators on a Banach space form a unital Banach algebra under composition and the operator norm. If $K$ is compact Hausdorff, the continuous complex-valued functions on $K$ , with pointwise multiplication and the supremum norm, form a commutative unital Banach algebra. Given a Haar measure on a locally compact group, the integrable functions form a generally noncommutative Banach algebra under convolution.

Conventions and scope

Some authors build unitality into “Banach algebra”; others, as here, do not. Without the normalization $\lVert1\rVert=1$ , submultiplicativity only forces $\lVert1\rVert\geq1$ . A normed algebra need not be a Banach algebra unless it is complete. A $C^*$ -algebra is a complex Banach algebra with an involution satisfying the additional $C^*$ -identity, not merely a Banach algebra carrying some involution.

References

F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. Springer DOI record. Relevant: “Concepts and Elementary Results.”