Definition of C$^\ast$-algebra: which conditions can be deduced from the others?

Question

A C$^\ast$-algebra is a Banach algebra $A$ with an involution, i.e. a map $\ast$ such that:

1. $(x^\ast)^\ast=x$ for all $x\in A$;
2. $(x+y)^\ast=x^\ast+y^\ast$ for all $x,y\in A$;
3. $(ax)^\ast=\overline ax^\ast$, for all $x\in A,a\in\mathbb{C}$;
4. $\|xx^\ast\|=\|x\|^2$, for all $x\in A$.

I know 1-2-3 do not imply 4 (counterexample: $L^1(\mathbb{R})$ with convolution and $f^\ast(x)=\overline{f(-x)}$ and $\ell^1(\mathbb{Z})$ with convolution and $f^\ast(k)=\overline{f(-k)}$, and also Wiener algebra with whatever matches the involution on $\ell^1$ via the algebra isomorphism sending an element of Wiener into its FOurier transform in $\ell^1$, conjugation perhaps?), but is there any of 1-2-3 that can be deduced by the other conditions? Or more explicitly:

1. Can 1 be deduced from 2-3-4?
2. Can 2 be deduced from 1-3-4?
3. Can 3 be deduced from 1-2-4?

And how is any of the above 3 implications proven/disproven?

Answer 1

Your definition of $C^*$-algebra is missing the property $(xy)^*=y^*x^*$. Without that requirement, a more natural (to me) example where 1-2-3 are true and 4 is false is to take $M_2(\mathbb C)$ and define the adjoint as conjugation entrywise.

1. Condition 1 cannot be deduced from 2-3-4. For example, let $A=\ell^\infty(\mathbb N)$, and let $$x^*=(\overline{\gamma(x)},\overline{x_1},\overline{x_2},\ldots),$$ where $\gamma$ is any state.
2. I don't think that condition 2 can be deduced from 1-3-4. But I cannot think of an example right now.
3. Condition 3 cannot be deduced from 1-2-4. For example, take $A=\mathbb C$, and the identity as involution.