Two inequalities in $C^*$ algebras

Question

Under what conditions on a $C^*$ algebra $A$ we have the following inequality:

$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$

The second identity which I am looking for is the following:

Does the following inequality imply that the algebra is commutative:

$$xx^*\leq k x^*x\;\;\forall x\in A$$ for some positive real $k$? What about if we assume that $k$ is a fixed positive elemnt of the algebra rather than a positive scalar?

The second question is motivated by the fact that every Banach algebra which satisfies $|ab|\leq k|ba|,\;\;\forall a,b$ is necessarily a commutative algebra. The proof is based on Liouville theorem. But in our case the conjucation arising from $*$ operation destroy holomorphicity.

Answer 1

The second condition also implies that $A$ is commutative. If $A$ is not commutative then it has an irreducible representation on some Hilbert space $H$ of dimension at least $2$. Find unit vectors $v,w \in H$ with $\langle v, w\rangle = 0$. By Kadison transitivity there exists $x \in A$ with $xv = 0$ and $xw = v$. Then $\langle x^*xv, v\rangle = \|xv\|^2 = 0$ but $\langle xx^*v, v\rangle = \|x^*v\|^2 \neq 0$ because $\langle x^*v, w\rangle = \langle v, xw\rangle = 1$. So $xx^* \leq kx^*x$ is impossible.

(The second part of the question doesn't really make sense because if $k \in A$ then $kx^*x$ will not be positive in general. You could ask about $xx^* \leq x^*kx$, but this would imply $xx^* \leq \|k\|x^*x$ and therefore $A$ must be commutative by the scalar case.)

Answer 2

As observed, the quadratic term may be equivalently removed from the inequality due to different homogeneity; then $x^*a^*ax+a^*x^*xa\leq a^*x^*ax+x^*a^*xa$ can be rewritten $[a,x]^*[a,x]\le0$, so the condition is exactly: commutativity of $A$.