Tensor subalgebras and First Fundamental Theorems in invariant theory

Let $V={\text{oC}}^n$ and let $T:=T(V)\otimes T(V^{\ast })$ be the mixed tensor algebra over $V$. We characterize those subsets $A$ of $T$ for which there is a subgroup $G$ of the unitary group $\text{UU}(n)$ such that $A=T^G$. They are precisely the nondegenerate contraction-closed graded $\ast$-subalgebras of $T$. While the proof makes use of the First Fundamental Theorem for $\text{GL}(n,\text{oC})$ (in the sense of Weyl), the characterization has as direct consequences First Fundamental Theorems for several subgroups of $\text{GL}(n,\text{oC})$. Moreover, a Galois connection between linear algebraic $\ast$-subgroups of $\text{GL}(n,\text{oC})$ and nondegenerate contraction-closed $\ast$-subalgebras of $T$ is derived.