Let $G$ is a locally compact group (non-Abelian)


Why $sp(L^1(G))$ , i.e. the set of all nonzero bounded multiplicative functionals on $L^1(G)$ is a locally compact group.


Even for any noncommutative Banach algebra A, why $sp(A)$ is a locally compact space?


Can you give me a reference?



spectrum of Banach algebras
If $A$ is a (noncommutative) Banach algebra, a character (i.e. a multiplicative linear functional) is automatically bounded. Then consider the set $\Sigma(A) \subseteq A^\*$ of nonzero characters with the weak-$\*$ topology inherited from $A^\*$. Then $\Sigma(A) \cup \{ 0 \} $ is a weak-$\*$ closed subset of the unit ball in $A^\*$, which itself is weak-\* compact. Thus $\Sigma(A) \cup \{ 0 \} $ is weak-$\*$ compact, so $\Sigma(A)$ is locally compact.


Of course, for a general Banach algebra you may not have many characters, but that's a separate issue...


