### 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

Thus .

[…] Proof of Corollary 3: With the notation of the functional calculus, we have that , where is the continuous function on given by . Then is the required square root (the function is just ; sorry for the pedantry!). The uniqueness is left as an exercise – you can find a solution at the end of this post. […]