Advanced Analysis, Notes 15: C*-algebras (square root)
by Orr Shalit
This post contains some make–up material for the course Advanced Analysis. It is a theorem about the positive square root of a positive element in a C*-algebra which does not appear in the text book we are using. My improvisation for this in class came out kakha–kakha, so here is the clarification.
Definition 1: A normal element in a unital C*–algebra is said to be positive, denoted , if .
Theorem 2: Let be a normal element in a unital C*–algebra . Then if and only if there exists a positive element such that . When this occurs, the element is unique, and is contained in the C*–algebra generated by and .
Remark: The element in the above theorem is referred to as the positive square root (or sometimes simply as the square root) of and is denoted or .
Proof: Suppose that . Let be the positive square root function defined on . because . Let be given by the continuous functional calculus. Then is normal (because the functional calculus is a *–isomorphism from onto ) and it has the same spectrum as does in , which is . Thus . By the functional calculus .
Conversely, if with , then is normal and satisfies .
Suppose now that , and let in satisfy . Let as in the first paragraph. We will prove that . Let be a sequence of polynomials converging uniformly to on , and define another sequence . Then because , we have or . It follows that
uniformly on . Therefore . On the other hand