Dedicated to Goldie Hawn, on her birthday
نویسنده
چکیده
Robin Harte Dedicated to Goldie Hawn, on her birthday Abstract We attempt to deconstruct the Arens-Royden Theorem. Suppose A is a Banach algebra (by default complex, with identity 1): we shall write 0.1 A−1 = {a ∈ A : 1 ∈ Aa∩aA} for the open subgroup of invertible elements, and A−1 0 for the connected component of the identity in A −1: it turns out ([8], [11] Theorem 7.11.4) that 0.2 A−1 0 = Exp(A) = {e1e2 . . . ek : k ∈ N, c ∈ A} coincides with the generalized exponentials, the subgroup generated by the exponentials. Exp(A) is open, relatively closed in A−1, connected and a normal subgroup: thus we can form the quotient group, 0.3 κ(A) = A−1/Exp(A), the abstract index group [4];[13] of A. Now we can state [2];[4];[6];[18];[20];[21];[22] 1. Theorem (Arens-Royden Mark I) If A is commutative then 1.1 κ(A) ∼= H(σ(A),Z),
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