2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/111520Let $V=\oC^n$ and let $T:=T(V)\otimes T(V^*)$ 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 $\UU(n)$ such that $A=T^G$. They are precisely the nondegenerate contraction-closed graded $*$-subalgebras of $T$. While the proof makes use of the First Fundamental Theorem for $\GL(n,\oC)$ (in the sense of Weyl), the characterization has as direct consequences First Fundamental Theorems for several subgroups of $\GL(n,\oC)$. Moreover, a Galois connection between linear algebraic $*$-subgroups of $\GL(n,\oC)$ and nondegenerate contraction-closed $*$-subalgebras of $T$ is derived.Representation TheoryRings and Algebras15A72; 20Gxxtext