What do I need to do so that names with spaces also work? Any ideas?

The problem you're having is on the command line. When you do

  mp3concat -s 700 `ls`
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the results of ls become the command line for mp3concat, but if ls encounters filenames with embedded spaces, they'll appear as separate tokens on the command line. By the time these tokens get into @ARGV, it's too late.

One approach to fixing this is to have mp3concat take directory names as arguments (either instead of or in addition to filenames). Then, within the code, you could do something like

  foreach my $arg ( @ARGV ) {
      doDir($arg), next if -d $arg;
      doFile($arg) if -f $arg;
  }
  ...
  sub doDir {
      my $dir = @_;
      opendir(D, $dir) or die "$dir: $!";
      foreach my $file ( grep { -f } readdir(D) ) {
          doFile($file);
      }
      closedir(D);
   }
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With a little more work, this can handle hierarchies, as well.

Another alternative is to use File::Find to traverse directories, looking for .mp3 files. If you search around the monastary, you'll find plenty of examples of using File::Find.