# A property of a ideal of Banach algebras

Let $B$ be a Banach algebra and $A$ be a bi-ideal of it. Suppose that for any $b\in B$, $Ab=\{0\}$ implies $b=0$. Now could we say that for some $c\in B$ if $cA=\{0\}$ then $c=0?$

This definition is from "Banach Algebras and Automatic Continuity- H. G. Dales".

Definition 1.4.5 Let $A$ be an ideal in an algebra $B$. Then $B$ is left faithful over $A$ if $\{ b\in B : bA = 0\} = 0$, right faithful over $A$ if $\{b\in B : Ab = 0\} = 0$, and faithful over $A$ if $\{b \in B : bA=Ab=0\}=0$. An algebra $A$ is [left], [right] faithful if it is [left], [right] faithful over itself.

If I change your question face by this definition, we'll have

Let $B$ be a Banach algebra and $A$ be a bi-ideal of it, then $B$ is left faithful over $A$ iff it is right faithful over $A$.

I hope this can help!

• I'm a bit perplexed as to how you get "if and only if" here. There isn't any connection between the left/right pieces in the definition but maybe I'm dense. – Cameron Williams Apr 27 '15 at 14:14
• I'm quite confused on how this can help. – Crostul Apr 27 '15 at 14:18
• Thanks! But it doesn't help! – Hamid Shafie Asl Apr 27 '15 at 17:00