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In the early 1970s Pelczynski noticed that the only surjective isometries on $C(K)$ for the following compact Hausdorff space $K$ are $\pm Id$. I believe this was the first such example.

Quoting from Davis in "Separable Banach spaces with only trivial isometries": Let $K$ be a dendrite containing for each $n>2$ exactly one cut-point of degree $n$ so that the cut-points are dense in $K$. Then the only surjective homomorphism of $K$ is the identity." Banach-Stone then yields the result.

My question is: How does one construct $K$?

One of my issues is that I don't know some of the basic definitions. What is the degree of a cut point? Googling has not been very helpful.

Kevin Beanland
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    The degree of a cut-point is probably the number of components it separates the space into upon deletion. – mme Sep 06 '21 at 16:58
  • @mme. Thanks, that was my guess. – Kevin Beanland Sep 06 '21 at 17:05
  • Start with a closed interval of length $1$. Make the center $x$ of the interval a cut point of degree 3 by attaching an interval of length $1/3$ with end point at $x$. In each component of the three intervals you have when you remove $x$, make the center $y$ a cut point of degree $4$ by attaching two intervals of length $1/4$ with end points at $y$. Et cetera. – Bill Johnson Sep 06 '21 at 18:06
  • Bill: Does your construction give three cut points of degree 4 or am I reading it wrong? We need a unique cut-point for each degree. – Kevin Beanland Sep 06 '21 at 18:54
  • @Kevin at each stage you make the middle point of any pair of cut point a cut point, but whenever you add a new cut point you increase the degree by 1 by adding more new segments. This is all well defined because at every stage you are adding finitely many things. (If you don't want to have to worry about constructing this as a planar continuum which requires some care in making the segments shorter and shorter just build it as a subdendrite of Wazewski universal dendrite $D_\infty$) – Alessandro Codenotti Sep 06 '21 at 20:22
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    See also the second answer here https://mathoverflow.net/questions/188707/hausdorff-spaces-with-trivial-automorphism-group – Alessandro Codenotti Sep 06 '21 at 20:24
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    What Alessandro said. – Bill Johnson Sep 07 '21 at 03:14
  • Thanks everyone. I got it. – Kevin Beanland Sep 07 '21 at 13:12

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