High item-item internal homogeneity (what Alpha is mostly about) is not necessary for the items to be able to represent a latent factor validly (in the sense: unbiasedly).
Here is three variables data, V1, V2, V3, 50 cases. In one dataset (first 3 columns), correlations between all the three are 0.8. In the other dataset (next three columns) correlations between all the three are 0.3. The variables are standardized.
v1_0.8 v2_0.8 v3_0.8 v1_0.3 v2_0.3 v3_0.3 fsc_0.8 fsc_0.3
-.0216322 -.3145886 .1604344 .0233404 -.5238589 .3650942 -.0604708 -.0463575
.9333728 2.3119911 1.9546955 -.0931651 2.4597177 1.7832511 1.7888294 1.4205299
.3052711 1.0415126 -.6526027 .3129150 1.6869127 -1.4835223 .2387995 .1767378
.5030737 1.1074983 .0701299 .3364104 1.4587923 -.4845117 .5781644 .4486660
1.2797719 1.1299271 1.4849064 1.0008518 .7009884 1.3591254 1.3397510 1.0478070
2.2660289 2.7770475 2.0257540 1.7062525 2.6268757 1.2105131 2.4316898 1.8976580
-.2145210 -.1405609 -.2963220 -.1686681 -.0270311 -.3174341 -.2240841 -.1756520
-.5332263 -.4872407 -1.1308836 -.2352714 -.1383810 -1.3392088 -.7400683 -.5863339
-.4490196 -.3887868 -1.2903304 -.0895620 .0339002 -1.6494411 -.7320827 -.5836781
1.0338865 .6809572 .5711352 1.1060618 .4344277 .2254950 .7863807 .6045187
-.4836489 -.9900844 -.6503464 -.1494410 -1.0862012 -.4473419 -.7306871 -.5761066
-1.9360488 -2.1891431 -1.6425752 -1.5526669 -1.9972980 -.9659407 -1.9841218 -1.5458512
-1.2570528 -1.8665392 -1.3973651 -.7377985 -1.8553405 -.9706582 -1.5552170 -1.2199325
-.0546157 -.0376983 -.4180892 .0754412 .1096984 -.6011515 -.1755796 -.1424061
-1.3231616 -.2141172 -.9050470 -1.5870724 .4998681 -.7890407 -.8401643 -.6422622
.0691481 -1.0107037 -1.7590763 1.0631309 -.9432335 -2.3390808 -.9290220 -.7596543
.1556118 .3593251 -.1902539 .1709739 .5504975 -.4781568 .1116915 .0832896
1.2228620 1.1568216 1.5301791 .8909594 .7477838 1.4402742 1.3449995 1.0539863
-2.5730333 -1.7875258 -1.4853883 -2.6979986 -1.1993673 -.6252286 -2.0110160 -1.5481409
1.9842442 1.1221856 1.7050726 1.9785803 .3418057 1.4249476 1.6551651 1.2820747
.9770592 -.0441422 .8725782 1.1729472 -.7465142 .9657704 .6210934 .4765687
-1.2069782 -.3847021 -.1605063 -1.6095356 -.0626192 .3594374 -.6027552 -.4493597
-.7172861 -.8802134 -1.0844749 -.3874150 -.6787273 -1.0567414 -.9226038 -.7266897
-.2682062 .3626536 -.1105669 -.4896474 .6906040 -.1947034 -.0055451 .0021405
1.6929962 1.2566269 1.2078208 1.6677241 .8306217 .7329766 1.4301679 1.1061222
-1.4159937 -1.1830667 -.6379551 -1.4762627 -1.0244063 .0003235 -1.1135389 -.8558996
-.1500775 -.6646840 -.0972387 .0435106 -.9146387 .1483573 -.3137296 -.2474135
-.1828678 -1.1630316 -.5886179 .3388003 -1.4851117 -.4074865 -.6654773 -.5318845
.6590666 .4445127 .6415163 .6060979 .1959781 .5618709 .6003159 .4668962
1.5162195 .6415474 2.0699336 1.3243912 -.3331751 2.3326103 1.4543363 1.1377875
.2245677 -.0234253 .9045059 .0324350 -.4371384 1.2971452 .3803449 .3054940
-.0926322 -.0904877 .1399430 -.1543786 -.1501836 .2809690 -.0148529 -.0080762
.3780963 -.2112872 -.1955229 .7054150 -.3969598 -.3673892 -.0098776 -.0201739
-.4290316 -1.1542862 -.9926653 .1074007 -1.2363992 -.9300543 -.8861426 -.7048396
-.7585939 -.2935150 -.0498380 -1.0109333 -.1354595 .3220680 -.3790716 -.2821767
-.0843910 .1888064 .0036812 -.1935725 .3169627 -.0295486 .0371854 .0321232
.8418637 .4953959 -.2870958 1.1798177 .5265572 -.9389054 .3612582 .2627144
-.4998451 .4540655 .1387108 -.9529158 .8310741 .2409140 .0319685 .0407599
.2479741 -.0448949 .8637955 .0888841 -.4644383 1.2339117 .3670067 .2938266
-.7611045 -1.1167585 -1.5739342 -.2013707 -.8493105 -1.6992832 -1.1874242 -.9413473
.0660042 .3296798 .5710234 -.2137543 .2746073 .7246142 .3325490 .2688753
-1.4107885 -1.7275688 -.9734133 -1.1616271 -1.7337159 -.3165378 -1.4144562 -1.0994671
-.1167343 .4900621 -.7025693 -.1052139 1.0316952 -1.1989958 -.1132596 -.0932851
-.2279270 -.4127442 .4400241 -.3458270 -.6906678 .9049981 -.0690230 -.0450129
.1772902 .2251722 .1172732 .1448735 .2318547 .0292001 .1787899 .1389544
1.2545414 .6914799 .5711250 1.4314428 .3655735 .1365945 .8659027 .6618993
-1.3840245 .0592204 -.1973831 -2.0171455 .6902884 .2124831 -.5236350 -.3814642
1.0357746 .2783454 .9829308 1.1091084 -.3193657 .9952914 .7901894 .6110396
.0793607 1.1721525 .8984689 -.6028225 1.4306523 .9152923 .7395975 .5966926
-.3516433 .0448097 -.4555760 -.3837007 .3618054 -.5731659 -.2622702 -.2036969
Both the "0.8" set and the "0.3" set were generated from the same random normal values "ingot" by method described here; because of that, the two obtained bunches of variables-vectors, "0.8" bunch and "0.3" bunch, are oriented in the space of individuals (cases) the same way and the only difference between the two is the angles between the vectors: the "0.8" bunch is more tight vectors and the "0.3" bunch is more loose vectors.
Cronbach's alpha in set "0.8" is .923 and in set "0.3" is .563. It is clear that "internal consistency" (if to let Alpha personify that phrase) is much higher there where correlations are bigger then where correlations are smaller. Alpha is also dependent on the number of items in the construct, but here we have equal number of variables in both sets.
Let us assume that the construct uniting the three items for us is representing a latent factor, i.e. factor-based construct. Then factor analysis should support the construct. Perform factor analysis extracting one factor.
Loadings obtained in FA of "0.8" set
LOAD
.8944044699
.8944044699
.8944044699
They restore correlations ideally
LOAD*T(LOAD) [commulalities on the diagonal]
.7999593558 .7999593558 .7999593558
.7999593558 .7999593558 .7999593558
.7999593558 .7999593558 .7999593558
Loadings obtained in FA of "0.3" set
LOAD
.5477000812
.5477000812
.5477000812
They restore correlations ideally
LOAD*T(LOAD) [commulalities on the diagonal]
.2999753790 .2999753790 .2999753790
.2999753790 .2999753790 .2999753790
.2999753790 .2999753790 .2999753790
You see that in both cases FA extracted a factor which successfully explained the observed correlations (and thus the validity of the factor was found [I don't say it was confirmed, since we do EFA, not confirmatory FA nor cross-validation]). However, loadings (which are the correlations of a factor with items) were higher in "0.8" data than in "0.3" data. In other words, items in "0.3" set are only weakly driven by the common factor; they are still considerably individualized ("unique") and that's why correlations among them are as low as 0.3.
Let's compute factor scores of the factor in both analyses (see data columns fsc_0.8 and fsc_0.3 above). Here we used Regression method, but other methods, such as Anderson-Rubin, might be used as well. Because both data sets were generated as proportionally identical (except the strength of the correlations) in the space of individuals, factor scores from the two factor analyses correlate almost perfectly. In other words, we have yet another evidence of having the same latent factor operating in set "0.8" and set "0.3"; the only difference being between the sets that the factor loads items strongly in "0.8" and weaker in "0.3". So, variables in both our sets are equally valid to represent the factor in the sense that they belong to the same factor without any bias, but in one set they belong to the factor much and in the other set they belong to the factor little. Since factor scores fsc_0.8 and fsc_0.3 are linearly identical, it's no difference which of them you input to, say, a regression analysis as a dependent variable.
Of course, items which belong together much are closer to be duplicates of each other and will have higher Alpha internal tightness. But, as we see, low alpha is still compatible with good validity in the sense of factor inbiasedness. Validity is not the same as reliability (1, 2).
Moreover, if we turn to speak away from construct validity towards external validity, i.e. the correlation of the factor (e.g. of factor scores or of sum of items) with some outside criterion variable embodying the trait, we might find that it is not necessarily so that the correlation will be higher in set "0.8" than in set "0.3".
Why would we, despite all said, prefer to have data "0.8" than "0.3"? Because if we add random noise (i.e. add measurement error) to the variables, correlations in "0.8" will drop to, say, 0.6, which is enough eligible for a factor analysis to show good results in real settings, whereas "0.3" will drop to 0.1 which is already too low correlations to extract the factor mathematically consistent (in real data settings). It is more difficult, having the same detection apparatus, to find a small needle in a vast haystack than an awl in a tight haystack.
