= Identity component of Banach-algebra invertibles
{title2=$G_0(A)=\langle e^a:a\in A\rangle$}
Finite products of <Banach algebra exponentials> form a <subgroup>: inverses are $e^{-a}$ and conjugation sends $e^a$ to $e^{gag^{-1}}$. Each product is joined to the identity by $t\mapsto\prod e^{ta_i}$. The <logarithm of an element near the identity> puts a <norm> neighborhood of $1$ in this <subgroup>, so its cosets are <open> and its complement is <open>. It is therefore exactly the <identity component of a topological group>, and is an <open>, <closed>, <normal subgroup>. A product of exponentials need not be a single exponential in a noncommutative algebra.
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