Ok I have finally migrated things over to a GitHub.io repo and now things are infinitely easier on my end. So as a sort of "test-drive," I will walk through the basic classification of operators in $C^*$-algebras with respect to the given involution. I will also give some statements of fundamental theorems in operator algebra theory.

Definition

Def: Let $A$ be a Banach algebra over a field $\mathbb{F}$ with an involution $a \mapsto a^*$. Then $A$ is a $C^*$-algebra.

Note that I didn't really mention what the involution does or the requirements therein. I will gloss over those for now but keep in mind there are certain properties required by the involution. The only one to be concerned about for now is the $C^*$-identity $\|aa^*\| = \|a^*a\|$.

For an ideal $I \subset A$, the quotient algebra $A/I = \{a+I \mid a \in A \}$ is also a $C^*$-algebra with norm $\| a+I\| = \inf \{ \| a+x \| \mid x \in I\}$. Recall also that an algebra is simple if its only ideals are trivial.

Theorem (Gelfand-Naimark)

Thm: Gelfand-Naimark (Gelfand duality) Every commutative $C^*$-algebra is isometrically isomorphic to an algebra $C_0(X)$ for a locally compact Hausdorff space $X$.

As usual, $C_0(X)$ is the algebra of compactly supported functions on $X$. Further, every commutative $C^*$-algebra is isometrically isomorphic to a closed $C^*$-subalgebra of some $\mathcal{B}(H)$, the algebra of bounded linear operators on a Hilbert space $H$.

The Gelfand duality theorem is hugely important. Most $C^*$-algebras are not necessarily "nice" to work with, but compactly supported functions are very nice to work with. Gelfand duality gives us a robust way to bounce between the two. Note also that we get an isometric isomorphism, so even the metric and norms are equivalent in some way. The theory of compactly supported functions is extensive and runs very deep, so identifying the right isomorphism allows us to work in $C^*$-algebras as if we were working in a more familiar function algebra.

I could spew operator algebra theorems all day, but my intent here is really just to classify the most common/useful classes of operators in $C^*$-algebras. First, let's look at the spectrum of an operator and some neat properties.

Definition (Spectrum)

Def: Let $A$ be a $C^*$-algebra. Then for some $a \in A$, the spectrum of $a$ is the set

The spectrum of an operator enjoys lots of nice properties:

• $A = \{0\} \implies \operatorname{sp}_A(0) = \varnothing$
• $\operatorname{sp}_A(\lambda 1_A) = \{\lambda\}$
• $a$ is invertible iff $0 \notin \operatorname{sp}_A(a)$
• For a complex polynomial $P$, $\operatorname{sp}_A(P(a)) = P(\operatorname{sp}_A(a))$
• If $a$ is nilpotent then $\operatorname{sp}_A(a) = \{0\}$
• If $\varphi:A \to B$ is a morphism of complex unital algebras, then $\operatorname{sp}_B(\varphi(a)) \subseteq \operatorname{sp}_A(a)$
• If $(a,b) \in A \oplus B$, then $\operatorname{sp}_{A \oplus B}((a,b)) = \operatorname{sp}_A(a) \cup \operatorname{sp}_B(b)$

Studying the spectra of operators is once again a very deep and interesting field, but beyond the context of this entry. Now, the basic classification of operators.

Let $a \in A$ be an operator in a $C^*$-algebra $A$, then we can classify $a$ as:

We denote by $A^+$ the set of all positive operators in $A$.

There are many more classifications of operators (trace class, Hilbert-Schmidt, etc.), but the definitions are a little more involved. You will not be able to read a single paper in functional analysis or operator algebras without bumping into the operator classifications given above. These are super important definitions, and the properties of each classification could each easily populate their own entry.

I don't have much more today — this is mostly to get some kinks worked out with Git and formatting, but I figured I may as well mention something foundational in operator theory and, ultimately, noncommutative geometry.