This might be a bit of a stupid question but I'm working on a question of Kirillov's "Elements of the Theory of Representations", where he proves that you can decompose a unitary representation of a locally compact group into the direct integral of irreducible representations.
I can understand most of it just fine, but I'm really stuck on figuring out what "the isomorphism from H to $\int_X H_x d\mu_X$ which maps the vector $T(a)\xi \in H$ to the vector function $\xi(x) = T_x(a)\xi_x$" means.
$T_x$ are the set of representations onto some Hilbert spaces $H_x$, which is 'indexed' by a certain set X – I can give more context if needed, but I don't think too many details of the question are relevant. I just don't understand what the actual isomorphism is.
It suggests taking some $\xi_x \in H_x$ but then surely that just fixes the same result $T_x(a)\xi_x$ for any $\xi$? Likewise, it seems to suggest the input is $x \in X$ but then that's not an isomorphism from H.
I think perhaps it could be forming a map from H to each $H_x$ instead.
Again, this is a bit of a dumb question but I can't figure it out for the life of me. I know how to proceed once I've worked out that isomorphism.
Best Answer
Let $H_x$ be the family of Hilbert spaces indexed by $x\in X,$ and $\mu_X$ the measure on $X.$ Let $H_X$ be the Hilbert space consisting of families of vectors $\xi=\{\xi_x\},$ $\xi_x\in H_x,$ such that $$\|\xi\|_{H_X}^2=\int\limits_X\|\xi_x\|^2_{H_x}\,d\mu_X(x) <\infty$$ The inner product in $H_X$ is then well defined as $$\langle \xi,\eta\rangle_{H_X} =\int\limits_X\langle \xi_x,\eta_x\rangle_{H_x}\,d\mu_X(x)$$
The theorem states: for every representation $G\ni a\to T(a)\in B(H)$ there exists a Hilbert space of the form $H_X,$ and an isometric isomorphism $U:H_X\to H$ such that the representation $\tilde{T}(a):=U^{-1}T(a)U$ (equivalent to $T$), acting on $H_X,$ is of the form $$\tilde{T}(a)\xi =\int\limits_XT_x(a)\xi_x\,d\mu_X(x)\qquad (*)$$ where $T_x$ are inequivalent irreducible representations of the group. The equality $(*)$ is defined in the weak sense, i.e. as $$\displaylines{\langle\tilde{T}(a)\xi,\eta\rangle_{H_X} \\ =\int\limits_X\langle T_x(a)\xi_x,\eta_x\rangle_{H_x}\,d\mu_X(x)}$$
Remark If the representation $T$ is finite dimensional then $X$ is a finite set, and the measure $\mu_X$ is determined by the multiplicity of a given irreducible subrepresentation $T_x$ in $\tilde{T},$ i.e. by equivalence in $T.$