Is every theorem about unital Banach algebra also true for non-unital Banach algebras because of unitization?
2026-03-26 02:58:11.1774493891
Unitization of Banach algebras
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Of course not. For example the Gelfand spectrum of Banach algebra is locally compact, and it is compact iff $A$ is unital. There a lot of examples in Banach homology where existence of unit plays significant role.
Another example which is more related to Banach space geometry than to Banach algebras: the Banach algebra $c_0$ have no extreme points while its unitization $c$ have $\mathfrak{c}$ extreme points.