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Just trying to make out if AVE is less than .5 but CR and Cronbach Alpha is v. good what should be done?
For example AVE= .398 ; Composite Reliability=.867 ; Cronbach Alpha=.829 ; rho_A=.840
I know recommended AVE is not less than .5

However, after bootstrapping the results I have got are also significant.

What should be done?

First and foremost AVE and reliability are similar but two different concepts.
Second, some of the reliability criteria improve if you have many indicators (e.g., Cronbachs alpha). Thus, many medium indicator give good Cronbachs Alpha, but weak AVE.
Third, the composite reliability criterion usually overestimates the true reliability, so it is not a good indicator anyway.

It is hard to tell more without knowing the exact results.

Thank you so much prof Becker.
With the deletion of one indicator with low loading the minimum AVE .5 could be achieved.