Topological properties in tensor products of Banach spaces

Given two Banach spaces X and Y, we analyze when the projective tensor product X?ˆ?Y has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a compactly generated (WCG) space Hilbert space. For instance, show that: (i) WLD if only both Y are all operators from to Y? X? have separable range; (ii) WCG same holds for under assumption that every operator compact; (iii) ?p(?)?ˆ??q(?) any 1<p,q<? such 1/p+1/q<1 infinite set ?. We also pay attention injective X?ˆ?Y. In this case, stability being turn out be closely related condition regular Borel probability measures on dual ball countable Maharam type. Along way, generalize result Plebanek Sobota K compact C(K×K) (C), then This generalization provides consistent negative answer question Ruess Werner about preservation w?-angelicity unit products.