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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?

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$L^1(G)$ 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.

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    This should be in the book somewhere...2012-09-24
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    Maximal ideals are already closed!2012-09-24
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    Right. Let's say it was for emphasis :-)2012-09-24
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    He defines the spectrum of an operator as the set of eigenvalues. On a commutative unital Banach algebra he defines it as the set of multiplicative functionals on the algebra. I guess this might be related to your definition.2012-09-24
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    A multiplicative functional is determined by its kernel, which is a (closed!) maximal ideal of the algebra. This is done in section 1.2 of the 1st chapter of the book.2012-09-24
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    So .. maximal ideals are closed even in a Banach algebra without unit...2012-09-24