If $G$ is a locally compact abelian group, what does "the spectrum of $L^1(G)$ mean?" This comes from Folland's A Course in Abstract Harmonic Analysis. As I understand it, $L^1(G)$ is the integrable functions (wrt Haar measure) on G. I always think of spectrum as the set of eigenvalues of an operator (or I guess the set of prime ideals if we're taking the spectrum of a ring). Can someone clarify for me the situation? Folland tells us on page 88 that the spectrum of $L^1(G)$ can be identified with the P-dual of $G$. What is going on?
$L^1(G)$ is a commutative Banach algebra. As such, it has a spectrum: as a set, this is the set of its closed maximal ideals. It is also a topological space in a certain canonical way.