What is the $C^*$-algebra generated by a normal operator?

Question

The following is the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas:

I don't find the definition for the $C^*$-algebra generated by a normal operator in the book. Would anyone point me to a reference illustrating what it is? Also, in the context of matrices, say a $n\times n$ complex matrix $T$, what is the generated $C^*$-algebra $\mathfrak{C}_T$?

Answer 1

The $C^*$-algebra generated by a normal operator $T$ is the closure in $B(\mathscr H)$ (the bounded linear operators on $\mathscr H$) of the polynomials in $T$ and $T^*$. In the case of $n \times n$ matrices, since $T$ and $T^*$ satisfy polynomials of degree $n$ (namely their characteristic polynomials), you just need polynomials in $T$ and $T^*$ of degrees $\le n-1$ in both $T$ and $T^*$.

Answer 2

There's a more general notion of C*-algebras generated by subsets. Let $A$ be some C$^*$-algebra, unital or not. Suppose $F$ is any nonempty subset in $A$, then we can form the smallest C*-algebra generated by $F$, sometimes denoted $C^*(F)$, by taking all intersections of sub C*-algebras containing $F$ or more concretely: Collect all polynomials of elements in $F$ and $F^*$ where $F^*=\lbrace x^*: x\in F \rbrace$ and take the norm closure of this $*$-algebra to form a C$^*$-algebra. The reason why the theorem requires the element in question to be normal is to ensure that $C^*(\lbrace T \rbrace)$ becomes commutative.