The polarity classification task aims at automatically identifying whether a subjective text is positive or negative. When the target domain is different from those where a model was trained, we refer to a cross-domain setting. That setting usually implies the use of a domain adaptation method. In this work, we study the single and cross-domain polarity classification tasks from the string kernels perspective. Contrary to classical domain adaptation methods, which employ texts from both domains to detect pivot features, we do not use the target domain for training. Our approach detects the lexical peculiarities that characterise the text polarity and maps them into a domain independent space by means of kernel discriminant analysis. Experimental results show state-of-the-art performance in single and cross-domain polarity classification.