Let $H$ be a Hilbert space and $A\subset B(H)$ be a unital C*-algebra. If the identity operator $1_{B(H)}$ is contained in $A$, then $1_A$ must be $1_{B(H)}$. However, if the identity operator $1_{B(H)}$ is not contained in $A$, is there any relationship between $1_{A}$ and $1_{B(H)}$?
2026-04-07 16:52:53.1775580773
The unit of C*-subalgebra of $B(H)$
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You will always have that $1_A$ is an orthogonal projection. So $$ 1_A\leq 1_{B(H)}. $$