equivalence of definitions of $C^{*}$-algebra

In this entry, we will prove that the definitions of $C^{*}$ algebra given in themain entry are equivalent.

Theorem 1.

A Banach algebra $A$ with an antilinear involution $\mathrm{*}$ such that${\mathrm{\parallel }a\mathrm{\parallel }}^{\mathrm{2}}\mathrm{\le }\mathrm{\parallel }{a}^{\mathrm{*}}\mathit{}a\mathrm{\parallel }$ for all $a\mathrm{\in }A$ is a $C^{\mathrm{*}}$-algebra.

Proof.

It follows from the product inequality $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$ that

$${\parallel a\parallel }^2\le \parallel a^{*}a\parallel \le \parallel a^{*}\parallel \parallel a\parallel .$$

Therefore, $\parallel a\parallel \le \parallel a^{*}\parallel$. Putting $a^{*}$ for $a$, we also have $\parallel a^{*}\parallel \le \parallel a^{**}\parallel =\parallel a\parallel$. Thus, the involution is an isometry: $\parallel a\parallel =\parallel a^{*}\parallel$.So now,

$${\parallel a\parallel }^2\le \parallel a^{*}a\parallel \le {\parallel a\parallel }^2.$$

Hence, $\parallel a^{*}a\parallel ={\parallel a\parallel }^2$.∎

Theorem 2.

A Banach algebra $A$ with an antilinear involution $\mathrm{*}$ such that$\mathrm{\parallel }{a}^{\mathrm{*}}\mathit{}a\mathrm{\parallel }\mathrm{=}\mathrm{\parallel }{a}^{\mathrm{*}}\mathrm{\parallel }\mathit{}\mathrm{\parallel }a\mathrm{\parallel }$ is a $C^{\mathrm{*}}$-algebra.