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

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

Theorem 1.

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

Proof.

It follows from the product inequality $\|ab\|\leq\|a\|\|b\|$ that

 $\|a\|^{2}\leq\|a^{*}a\|\leq\|a^{*}\|\|a\|.$

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

 $\|a\|^{2}\leq\|a^{*}a\|\leq\|a\|^{2}.$

Hence, $\|a^{*}a\|=\|a\|^{2}$. ∎

Theorem 2.

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

Title equivalence of definitions of $C^{*}$-algebra EquivalenceOfDefinitionsOfCalgebra 2013-03-22 17:42:27 2013-03-22 17:42:27 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Theorem msc 46L05 HomomorphismsOfCAlgebrasAreContinuous CAlgebra