CCR algebra

Remarks

Clause 2 with $u = v = 0$ gives $W(0)^2 = W(0)$, and $W(0) W(v) = W(v) = W(v) W(0)$, so $W(0)$ is the unit of $\mathcal{A}$. Then clauses 1 and 2 give $W(v)^* W(v) = W(-v) W(v) = W(0)$, so each $W(v)$ is unitary. Such an algebra exists, and any two are isomorphic by a $*$-isomorphism that matches the elements $W(v)$; so one speaks of the CCR algebra $\mathrm{CCR}(V, \sigma)$.

Example: let $V = \R^2$ with $\sigma((a, b), (a', b')) = a b' - a' b$, and let $\hbar = 1$. On $L^2(\R)$ define $(W(a, b)\psi)(x) = e^{i b (x - a/2)}\, \psi(x - a)$, which is $e^{i(bQ - aP)}$ for the position $Q$ and momentum $P = -i\, d/dx$. A direct computation gives $W(a, b) W(a', b') = e^{i(a' b - a b')/2}\, W(a + a', b + b') = e^{-i\sigma((a,b),(a',b'))/2}\, W(a + a', b + b')$, and $W(a,b)^* = W(-a, -b)$. So the C*-algebra generated by these unitaries is the CCR algebra over $\R^2$; this is the Schrödinger representation of the Stone–von Neumann theorem.

Non-example: the relation $QP - PQ = iI$ cannot be realized by bounded operators $Q$ and $P$ on a Hilbert space. So the unbounded generators are not elements of any C*-algebra, and this is the reason for clauses 1 and 2.

The map $v \mapsto W(v)$ is not norm continuous: $\norm{W(u) - W(v)} = 2$ whenever $u \ne v$.