Do you just need to know the max value for each unique entry in column B, or do you actually need to select the features with the maximum values on your map?
If you just need an output table with headings
[B], [Max_of_A], you can open the attribute table, then right-click column B and choose
Summarize. In the list of summary statistics to be calculated, choose
Maximum for column A and you should end up with the output you need.
If you need to actually select the features on your map, one way would be to join the table produced by
Summarize back to your original features. Let me know if you'd like more details.
Someone else may have a better way, but I don't think you can do this directly using ArcMap's
Select by Attributes option(?). Would be interested to be proved wrong, though.
As @GIS-Jonathan says, if your data is stored in a spatial database and you're just using ArcMap to view results etc. then your best bet is to use a GROUP BY query as per his method. If you're working with a shapefile or a feature class stored in a file or personal geodatabase, the method below should work, but it's a bit of a faff! 300,000 records shouldn't be a problem, though.
Start by using
Summarize on your original table. You should get an output something like this:
Now join this table back to your original dataset using column B as the join field (right-click your original dataset, then
Joins and Relates > Join. Choose to keep all records). This should make your attribute table look something like this:
[Orig_A] [Orig_B] [Summary_B] [Max_of_A]
100 aaa aaa 999
101 aaa aaa 999
102 aaa aaa 999
999 aaa aaa 999
100 bbb bbb 999
101 bbb bbb 999
102 bbb bbb 999
999 bbb bbb 999
Then you can use
Select by Attributes to select all the rows where
[Orig_A] = [Max_of_A].
Quite a faff, but I can't think of an easier way to do it if your data isn't already in a spatial database supporting GROUP BY. Maybe someone else can suggest a better way?