Proposition 5

Let A be a unital star algebra and B a star subalgebra of A containing the unit element of A. Then B is a full subalgebra of A. In particular, one has Sp$_B(x) =$ Sp$_A(x)$ for every $x$ in B.

Let $x$ be a hermitian element of B. Since Sp$_B(x)\subset \mathbf{R}$ (Prop. 4), Prop. 6 of I, p. 28 shows that Sp$_B(x) =$ Sp$_A(x)$. In particular, $x$ is invertible in B if and only if it is invertible in A.

Let now $x$ be any element of B invertible in A. Then $x^*$ is invertible in A, and $xx^*$ is invertible in A. Since $xx^*$ is hermitian, what precedes shows that $xx^*$ is invertible in B. This implies that $x$ is right-invertible in B. Analogously, one verifies that $x$ is left-invertible in B, and hence that $x$ is invertible in B. Thus, B is a full subalgebra of A.