## Direct integral

People seem to really like replacing sums by integrals. It turns out that even something like a direct sum of Hilbert spaces can be generalized. The decomposition of the regular representation of the circle $\mathbb{T}$ (which is a Lie group) into irreducible unitary representations gives us the theory of Fourier series. One thing that confused me for a while is another thing called the “Fourier transform”, which is when you write a function on $\mathbb{R}$ (rather than a function on the circle/periodic function on $\mathbb{R}$) as an integral rather than a series. However, I realized it’s completely natural to do so: the analogous decomposition for the additive group of real numbers (which is also a Lie group) gives something a bit different: you can’t decompose $L^2(\mathbb{R})$ as an orthogonal direct sum like you did to $L^2(\mathbb{T})$, but you can decompose it into something called a “direct integral”…

Suppose you have a set (which will act as an “index set” for your spaces) $X$ equipped with a non-negative measure, $\mu$, and for each $x \in X$ you have an associated Hilbert space $\mathfrak{H}_x$. Suppose each $\mathfrak{H}_x$ is of the same dimension, so we can identify them all with one space $\mathfrak{H}$. We denote by $\mathcal{H}$ the space of functions $\xi : X \to \mathfrak{H}$ such that for any $v \in \mathfrak{H}$, the function $x \mapsto (\xi(x), v)$ is $\mu$-measurable, and the function $x \mapsto \| \xi(x) \|$ is square-integrable w.r.t. $\mu$, that is,

$\displaystyle \int_X \| \xi(x) \|^2 \; d\mu(x) < \infty.$

Then $\mathcal{H}$ becomes (as one can show) a Hilbert space under the obvious pointwise operations of vector addition and scalar multiplication, and the scalar product

$\displaystyle (\xi, \eta) = \int_X (\xi(x), \eta(x)) \; d \mu(x).$

$\mathcal{H}$ is called the continuous direct sum (or direct integral) of the spaces $\mathfrak{H}_x$ with respect to $\mu$, and we write it as

$\displaystyle \mathcal{H} =: \int_X^{\bigoplus} \mathfrak{H}_x \; d\mu(x).$

And that is where the fun begins: F.e. every unitary representation of a locally compact group on a Hilbert space is a direct integral of irreducible representations. The simplest example is probably the fact that by using the Fourier transform one has that the representation of $\mathbb{R}$ on $L^2(\mathbb R)$ is nothing else but the direct integral of copies of $\mathbb{R}$, or more precisely of the Hilbert spaces $H_y= e^{icy}$ on which $\mathbb{R}$ clearly acts.