I'm currently reading the book of Folland A Course in Abstract Harmonic Analysis and I'm stucked with the proof of the following proposition:
If $A$ is a nonunital $C^*$ algebra, there is a unique norm on $\tilde{A}$ that makes $\tilde{A}$ into a $C^*$ algebra with involution $(x,a)^*=(x^*,\overline{a}$). This norm agrees with the original norm on $A$.
I am now stucked in the part, where we wanna show that it's a norm: We have defined: $$\|(x,a)\|= \sup\{\|xy+ay\|: \;y \in A, \; \|y\| \leq 1\}$$ We suppose that $(x,a)$ is a nonzero element of $\tilde{A}$ satisfying $\|(x,a)\|=0$, so that $xy+ay=0$ for all $y \in A$. Clearly $x$ must be nonzero, and then $a$ must be nonzero since $xy\neq 0$ for $y=x^*$, so $z=-a^{-1}x$ is a left unit for $A$. But then $z^*$ is a right unit for $A$, so $z=zz^*=z^*$ is a unit for $A$, contrary by assumption.
I don't understand now why clearly $x$ must be nonzero. Is see no reason why we can assume this...