So how the amber color of LPS lamps considered as 1700K, despite it is monochromatic (Even more than an amber LED)?
Each black body temperature represents its spectrum. The "Correlated color temperature" is then obtained as the temperature of black body radiation with the highest correlation between the "tested" spectrum and the spectrum of a black body radiator at that temperature get the highest correlation product.
So even when the LPS is monochromatic, it yields some correlation product number with any other spectra, different with different ones. As the CCT parameter takes only those spectra emitted by a black body, spectrum of each compared temperature gives its corresponding correlation result. The temperature of the black body yielding the highest correlation is then defined as the CCT. So for the monochromatic LPS the highest correlation is obtained with the spectrum of a 1700K black body, hence the CCT=1700K.
The correlation is a mathematic operation between two functions
Correlation(func1, func2) = integrate(func1(x)*func2(x)*dx, for x within the definition range)
yielding a single number, telling how similar those functions are. It could be positive (very similar or identical functions), small number (dissimilar functions), zero (ortogonal functions), even negative (functions of opposite polarity values). Here those functions under question are power density as function of wavelength. Per definition the power density can be only positive, so the correlation can never be negative.
So when calculating correlation between two monochromatic lights, you get zero. When correlating any visible wavelength with a continuum over the whole visible range, you always get some nonzero value. And depend on the shape of the continuum spectrum, the correlation then could be higher (that wavelength is in the part where the continual spectrum has higher density) or lower (when the continual spectrum has lower density at the wavelength of the tested source). Of course the "reference" spectra of the black body radiator are normalized, so the correlation results could be compared.
The CCT then defines the wavelenght definition range to calculate the correlation as the "visible wavelength range", so the IR radiation of the black body is not taken into account.