Can 2 different language formats of tools without cross cultural validation be employed in 2 different populations, and CFA applied on the results to check invarience and establish divergent validity?
Also if one of these 2 language questionnaires shows factor loadings according to pre defined dimensions, while the other does not, can we still consider the one that loaded positively as validated ?
at first, some definitions to be sure we are on the same grounds:
Divergent validity: Participant's scores in target construct A show sufficiently low correlation with construct B, which is predicted by theory.
Convergent validity: Participant's scores in target construct A show sufficiently high correlation with construct A', which is predicted by theory.
It follows, that only constructs/latent variables can have convergent or divergent valdity, not items nor the measurement model. The concepts of divergent and convergent validity are independet from the underlying measurement models.
I wouldn't call a translation of a measure "divergent" when it yields evidence for noninvariance on some level (configural, metric or scalar).
If your are using the same items, then in order to consider them as "validated", the two groups should exhibit (a) good fit and the same, expected, loading patterns (i.e. configural invariance) and (b) the same factor loadings (i.e. metric invariance), and - if you are interested in comparing means - (c) the same item intercepts (i.e. scalar invariance). If these equalities cannot be obtained, then the two groups cannot be meaningfully compared.