Element of Unital Banach Algebra on Boundary of Group of Units of Subalgebra is Not Invertible in Algebra/Lemma

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Lemma

Let $\struct {A, \norm {\, \cdot \,} }$ be a unital Banach algebra.

Let $\map G A$ be the group of units of $A$.

Let $x \in \partial \map G A$, where $\partial \map G A$ is the topological boundary of $\map G A$.


Then there exists a sequence $\sequence {z_n}_{n \in \N}$ in $A$ such that $\norm {z_n} = 1$ for each $n \in \N$, and:

$z_n x \to 0$ as $n \to \infty$

and:

$x z_n \to 0$ as $n \to \infty$.


Proof

From Group of Units in Unital Banach Algebra is Open, we have:

$\partial \map G A = \map G A^- \setminus \map G A$

So if $x \in \partial \map G A$, then $x \in A \setminus \map G A$ and there exists a sequence $\sequence {x_n}_{n \in \N}$ with $x_n \in \map G A$ for each $n \in \N$, such that:

$x_n \to x$ as $n \to \infty$.

Let:

$\ds z_n = \frac {x_n^{-1} } {\norm {x_n^{-1} } }$

We have:

$\ds \norm {z_n} = \norm {\frac {x_n^{-1} } {\norm {x_n^{-1} } } } = \frac {\norm {x_n^{-1} } } {\norm {x_n^{-1} } } = 1$

from Norm Axiom $\text N 2$: Positive Homogeneity, for each $n \in \N$.

Now, we have:

\(\ds \norm {z_n x}\) \(=\) \(\ds \norm {z_n \paren {x - x_n} + x_n z_n}\)
\(\ds \) \(=\) \(\ds \norm {z_n \paren {x - x_n} } + \norm {x_n z_n}\) Norm Axiom $\text N 3$: Triangle Inequality
\(\ds \) \(\le\) \(\ds \norm {x - x_n} + \frac 1 {\norm {x_n^{-1} } }\) Definition of Algebra Norm, Definition of Unital Banach Algebra


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We know by hypothesis that $\norm {x - x_n} \to 0$, while from Norm of Inverse of Sequence of Invertible Elements Converging to Non-Invertible Element in Unital Banach Algebra, we have:

$\ds \frac 1 {\norm {x_n^{-1} } } \to 0$

So, we have:

$z_n x \to 0$ as $n \to \infty$.

Similarly, we have:

\(\ds \norm {x z_n}\) \(=\) \(\ds \norm {\paren {x - x_n} z_n + x_n z_n}\)
\(\ds \) \(\le\) \(\ds \norm {\paren {x - x_n} z_n} + \norm {x_n z_n}\) Norm Axiom $\text N 3$: Triangle Inequality
\(\ds \) \(\le\) \(\ds \norm {x - x_n} + \frac 1 {\norm {x_n^{-1} } }\) Definition of Algebra Norm, Definition of Unital Banach Algebra

So $\sequence {z_n}_{n \in \N}$ satisfies the conditions given in the theorem.

$\blacksquare$

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