Let $A$ be a pre-$C^*$-algebra, i.e. $A$ satisfies all axioms for a $C^*$-algebra except completeness. In other words, $A$ is an involutive algebra with a $C^*$-norm.

We say that $x \in A$ is positive (notation: $x \ge 0$) if there exists $a\in A$ with $x = a^*a$ and we define the usual partial order $x \le y \iff y-x \ge 0$.

Given a positive element $c \ge 0$ in $A$, and $a \in A$, does the inequality $$a^*ca \le \|c\| a^*a$$ hold? For $C^*$-algebras, this result is well-known. A quick proof is given by faithfully representing $A \subseteq B(H)$ and using $c \le \|c\|1$. Does the same result continue to hold for pre $C^*$-algebras?

Context question: In the book "Hilbert $C^*$-modules" by Lance on p4, it is claimed that all results proved so far continue to hold for inner product modules defined over a pre-$C^*$-algebra. The above inequality is used in the proof of proposition 1.1 and I wanted to check that it still works for pre $C^*$-algebras.

Let $A$ be a pre-$C^*$-algebra, i.e. $A$ satisfies all axioms for a $C^*$-algebra except completeness. In other words, $A$ is an involutive algebra with a $C^*$-norm.

We say that $x \in A$ is positive (notation: $x \ge 0$) if there exists $a\in A$ with $x = a^*a$ and on self-adjoint elements we define the usual relation $x \le y \iff y-x \ge 0$.

Given a positive element $c \ge 0$ in $A$, and $a \in A$, does the inequality $$a^*ca \le \|c\| a^*a$$ hold? For $C^*$-algebras, this result is well-known. A quick proof is given by faithfully representing $A \subseteq B(H)$ and using $c \le \|c\|1$. Does the same result continue to hold for pre $C^*$-algebras?

Context question: In the book "Hilbert $C^*$-modules" by Lance on p4, it is claimed that all results proved so far continue to hold for inner product modules defined over a pre-$C^*$-algebra. The above inequality is used in the proof of proposition 1.1 and I wanted to check that it still works for pre $C^*$-algebras.

Bonus question: Is the sum of two positive elements again positive?