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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 with $|ab|\leq k|ba|$ is necessarily commutative. the proof is based on Liouville theorem. But in our case the conjucation arising from $*$ operation destroy holomorphicity.

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.

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 with $|ab|\leq k|ba|$ is necessarily commutative. the proof is based on Liouville theorem. But in our case the conjucation arising from $*$ operation destroy holomorphicity.

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 with $|ab|\leq k|ba|$ is necessarily commutative. the proof is based on Liouville theorem. But in our case the conjucation arising from $*$ operation destroy holomorphicity.