What makes irreducible representations nice?

Question

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi,\Omega)$ a cyclic representation.

What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations are nice and I can be happy if my algebra can be represented in such a way, but which nice properties does an irreducible representation actually bring about?

Answer 1

I am teaching myself representation theory, and it seems to me irreducible representations are nice in two ways. (Since you are talking about representations of $C^*$-algebras, we might as well restrict our attention to $*$-representations, that is, the representation respects the involution.)

Extrinsically, irreps are nice just as prime numbers are nice since irreps are basic building blocks of general representations. Formally, this refers to

Every representation is a direct sum of irreducible representations.

So to study representations of an algebra $\mathcal{A}$, we might first try to find its irreps, and see how to break up a general representation into these blocks.

Irreps are also nice intrinsically. Since the algebra acts transitively on an irreducible representation, the geometry of an irrep is completely determined by this algebra. You might see its manifestations in the following

$\mathcal{H}$ is an irrep of $\mathcal{A}$ if and only if $\mathcal{A}\cdot v$ is dense in $\mathcal{H}$ for every $v\neq 0$.

Or indirectly, the von Neumann bicommutant theorem.

Well to conclude, irreps are nice because they are the smallest possible representations. The above two are just two faces of this smallness.

Answer 2

Hui Yu's answer gives some motivation for studying irreducible representations (irreps for short).

Here are few more reasons.

1.) A *-rep of a $C^*$-algebra, $\pi:A\rightarrow\mathcal{B(H)}$, is irreducible if and only if $\pi|_\Gamma$ is irreducible for any $\Gamma\subset\mathcal{U}(A)$ which is a subgroup of unitaries that generate A. So irreps of algebras are intrinsically linked to irreps of groups.

2.) Further for a compact group every representation is a direct sum of $\textit{finite dimensional}$ irreps. For more general locally compact groups the corresponding statement is that every representation is the direct integral of irreps (though the finite dimensionality no longer holds).

3.) The last one can be thought of as nice or not nice depending on your point of view. If $\pi:A\rightarrow\mathcal{B(H)}$ is irreducible then $\pi(A)$ is dense in $\mathcal{B(H)}$ in the weak (or strong) operator topology.

Answer 3

First of all, roughly every representation is a direct integral (as opposed to a direct sum) of irreducible representations. But this has to made precise. First of all, you can take the vNa-closure $N$ of the image and then $N$ decomposes into a direct integral (not necessarily unique) of factors. These correspond to irreducible representation of the algebra $N$ or extremal states on $N$. You really must argue with $N$ instead of the original $C^*$-algebra, I think.

Now, factors can roughly classified into three types. So in that sense, you obtain a coarse classification of irreducible representations of vNa.

Answer 4

In relation to physics: given a representation $\pi: \mathcal{A}\rightarrow \mathcal{B}(\mathcal{H})$, a vector $\mathbf{x}\in\mathcal{H} $ defines a "state" (positive normalized functional on $\mathcal{A}$) as $\omega_{\mathbf{x}}: \mathcal{A}\rightarrow \mathbb{C}, a \mapsto \frac{1}{||\mathbf{x}||^2} \langle \mathbf{x}|\pi(a)\mathbf{x} \rangle $ where $\langle \cdot|\cdot\rangle $ is the scalar product on $\mathcal{H}$. More states can be defined as so-called density matrices. The idea of this paragraph is that a representation gives a certain set of states on $\mathcal{H}$

However, from the abstract definition of states as positive normalized functional on $\mathcal{A}$ one can define pure and mixed states (as extremal points on the convex set of all states and points in the interior)

• For an irreducible representation $(\mathcal{H},\pi)$, the distinction pure/mixed and vector state/density matrices coincide. (To be complete I should also talk about the interpretation of "pure" as a "well-prepared" state, and mixed as a "incompletely prepared" state which I believe can be made mathematically clear but...)
• The set of vector states given by the data of an irreducible representation is only a subset of the set of all states. One can show that it corresponds to a connected subset of pure states

Comments @owen (I can't add any comments at the moment...): I'm very interested in possible links to groups, any reference? Also, I've seen that any von Neumann algebra can be obtained as the commutant of a (maybe continuous?) unitary representation of a group on a Hilbert space