Understand the spectrum of a C*-algebra

c-star-algebras

I know that the spectrum of an element $x$ in a unital C*-algebra $A$ is defined as
$$\operatorname{Sp}_{A} x=\left\{\lambda\in\mathbb{C}\mid (x-\lambda\cdot1)\ \text{is not invertible}\right\}.$$

Reference materials I am reading all seem to assume that the notion of spectrum of an element extends naturally to the notion of spectrum on a C*-algebra. Concretely, I am having issues understanding the idea behind the following two pieces of text.

If $A$ is commutative, then $A\cong C(X)$ for some compact Hausdorff space $X$, the spectrum is the range, and the spectral radius is the supremum norm.

Rieffel, M. 208 C^*-algebras, 2013

Let $A$ be a commutative C*-algebra, $S$ its spectrum (which is a locally compact space), and $B$ the C*-algebra of continuous complex-valued functions on $S$ which vanish at infinity. Then

 (i) Every character of $A$ is hermitian,
(ii) The Gelfand map is an isomorphism of the C*-algebra $A$ onto the C*-algebra $B$.

Dixmier, J. C*-Algebras, 1977

I might be missing some (trivial) connections, so here are the questions:

  1. What is the spectrum of a C*-algebra? How is it defined, and is it related to the spectrum of an element of a C*-algebra?

  2. Why is $B=C(S)$ involved, why not continuous complex-valued functions on some other set than $S$?

  3. If possible, how to intuitively understand:
    a. the spectrum $S$ of $A$,
    b. the isomorphism between $A$ and $C(S)$, which is also known as a Gelfand map?

Best Answer

The spectrum $\Omega(A)$ of a C$^*$-algebra $A$ is the set of characters, that is the nonzero $*$-homomorphisms $\varphi:A\to\mathbb C$. This set may be empty even in easy examples, like for instance when $A=M_n(\mathbb C)$, $n\geq2$.

When $A$ is commutative, though, we have the following nice result relating the two notions (spectrum of the algebra and spectrum of an element): for any $a\in A$, $$\tag{$*$} \sigma(a)=\{\varphi(a):\ \varphi\in\Omega(A)\}\cup Z_A, $$ where $Z_A=\varnothing$ if $A$ is unital, and $Z_A=\{0\}$ if $A$ is non-unital.

The relevance of the spectrum is not that much given by $(*)$, but rather by the Gelfand transform, which says that there is a natural isomorphism between the commutative C$^*$-algebra $A$ and $C(\Omega(A))$, where we consider the weak$^*$-topology in $\Omega(A)$. For the technicalities of the proof one uses the fact that there is a bijective correspondence between characters and maximal ideals via $\varphi\leftrightarrow\ker\varphi$.

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