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When I enter insert mode and type <C-SPACE>, vim inserts the last inserted text, then exits insert mode. My guess is it's using the ". register.
<C-V><C-SPACE> prints a null character. However when I enter the command :imap, it doesn't list any mappings. I was expecting there to be a mapping for the null character.
Can someone explain this behavior?
When you press <C-Space> in insert mode, Vim understands <C-@> which is a built-in command that inserts the previously inserted text and leaves insert mode.
You can't see <C-@> in :imap because it is not a mapping.
But you could have pieced all that together from the doc:
:help i_ctrl-a
:help ".
<C-Space>just puts a space for me even with<C-v>. Does it do this when starting vim withvim -u NONE?<space>to your leader? See the output of:imap <c-space>to find out if it is mapped to something. See also this question