K-projectors

We study representations of a free associative algebra $T^{\ast }(W\otimes W^{\ast })$ in a vector space $V$ with the property $V\otimes V\cong V\oplus V_0$ where $T^{\ast }(W\otimes W^{\ast })$ acts by zero on $V_0$ and the tensor product $V\otimes V$ of representations corresponds to the natural homomorphism $W\otimes W^\to W\otimes W^ \otimes W\otimes W^*$. We develop an algebraic theory of such objects and construct a lot of examples.