What is an example of a Banach algebra with a left identity but with no right identities? Is there an example of an operator algebra with this property?
2026-03-29 19:09:45.1774811385
One-sided identities in Banach algebras
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Consider the two-dimensional subalgebra of $M_2(\mathbb C)$ spanned by the matrices $$A := \begin{pmatrix}1 & 0 \\ 1 & 0\end{pmatrix} \text{ and } B := \begin{pmatrix}0 & 1 \\ 0 & 1\end{pmatrix}.$$ Then it is easy to verify that $A$ and $B$ both serve as left identities for this algebra. Obviously, being a subalgebra of a matrix algebra, this algebra can be given the structure of a Banach algebra and of an operator algebra.