I need help on proving the below question:
suppose that $A$ is a $C^{*}$-algebra and $\varphi:A\to\mathbb{C}$ is a linear, bounded map so that $\forall a \in A, \varphi ( a^{*}a ) = 0$.
Does this mean that $\varphi = 0$?
On
Ok, suppose you have some $a\in A$. Now, an element in $A$ is a sum of at most four positive elements, say $a = a_1 + a_2 + a_3 + a_4$. Now, write $a_i = b_i^*b_i$ for some $b_i \in A$ (each $a_i$ is positive). We have $\varphi(a) = \varphi(b_1^*b_1) + ... + \varphi(b_4^*b_4) = 0$ by linearity.
I would suggest the following. Any positive element $z$ in $A$ is of the form $z=a^*a$ for some $a\in A$ (non-trivial fact, some take it as a definition). Moreover, each element in a C$^\ast$-algebra is a linear combination of at most four positive elements (a consequence of the continuous functional calculus). Now, exploit linearity.