K-projectors

We study representations of a free associative algebra T⁎(W⊗W⁎) in a vector space V with the property V⊗V≅V⊕V0 where T⁎(W⊗W⁎) acts by zero on V0 and the tensor product V⊗V of representations corresponds to the natural homomorphism W⊗W⁎→W⊗W⁎⊗W⊗W⁎. We develop an algebraic theory of such objects and construct a lot of examples.