Let $A$ be a non-unital C*-algebra, and let $\pi: A\to B(A)$ defined by $\pi(a)(x)=ax$ be the left representation of A. Is the following inequality correct?

$$\lVert I+ \pi(a) \rVert\ge 1$$ for all $a in A$. ($I$ is the identity operator and $\lVert\cdot\rVert$ is the operator norm.)

If the inequality is correct, is it a known inequality?

Let $A$ be a non-unital C*-algebra, and let $\pi: A\to B(A)$ defined by $\pi(a)(x)=ax$ be the left representation of A. Is the following inequality correct?

$$\lVert I+ \pi(a) \rVert\ge 1$$ for all $a \in A$. ($I$ is the identity operator and $\lVert\cdot\rVert$ is the operator norm.)

If the inequality is correct, is it a known inequality?

Let A be a non-unital C*-algebra, and let π: A-->B(A) defined by π(a)(x)=ax be the left representation of A. Is the following inequality correct?

ǁ I+ π(a) ǁ>= 1 for all a in A. (I is the identity operator and norm is the operator norm)

If inequality is correct, is it a known inequality?

Let $A$ be a non-unital C*-algebra, and let $\pi: A\to B(A)$ defined by $\pi(a)(x)=ax$ be the left representation of A. Is the following inequality correct?

$$\lVert I+ \pi(a) \rVert\ge 1$$ for all $a in A$. ($I$ is the identity operator and $\lVert\cdot\rVert$ is the operator norm.)

If the inequality is correct, is it a known inequality?