# Where does the notation $\mathrm{Ad}(U)$ for $a\mapsto UaU^*$ come from?

I have often seen, in the context of operator theory and operator algebras, the notation $\mathrm{Ad}(U)a=UaU^*$, where $U$ is a unitary operator on a Hilbert space $H$ and $a$ is a bounded linear operator on $H$. I have no idea what "Ad" stands for, where/how this notation came into common use, nor whether it fits into a more general context (e.g., for similarities or other automorphisms outside of the context of operator theory). Some Google searching revealed a use of "$\mathrm{Ad}$" in the theory of Lie groups that doesn't quite match with the above, but might have a common origin.

Where does "$\mathrm{Ad}$" come from, especially in the context of $\mathrm{Ad}(U)a=UaU^*$?

• en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group – Qiaochu Yuan Apr 23 '11 at 18:27
• I'm pretty sure that @Qiaochu's right. I'd even go so far as to say that the motivation stems from the finite-dimensional case $A = M_{n}\mathbb{C}$, where the unitary group $U(n) = \mathcal{U}(A)$ leaves the subspace of self-adjoint matrices (= its Lie algebra) invariant. – t.b. Apr 23 '11 at 18:47
• Sorry, I should have said anti-self-adjoint ($a^{\ast} = - a$) before. – t.b. Apr 23 '11 at 19:01

I'm pretty sure that @Qiaochu's right. I'd even go so far as to say that the motivation stems from the finite-dimensional case $A=M_n(\mathbb C)$, where the unitary group $U(n)=\mathcal U(A)$ leaves the subspace of anti-self-adjoint matrices (= its Lie algebra) invariant. – t.b. Apr 23 '11 at 18:47