For a topological group G, we only want to consider unitary representations which are continuous in some sense. Since we have already seen above that the unitary group U(H) of a Hilbert space is a topological group with respect to the metric induced by the operator norm, it seems natural to call a unitary representationπ:G→U(H) continuous if it is continuous with respect to the norm topology on U(H). However, the norm topology on U(H) is very fine, so that continuity with respect to this topology is a condition which is much too strong for many applications. We therefore need a suitable weaker topology on the unitary group.
We start by defining some topologies on the space B(H) of all continuous operators which are weaker than the norm topology.
Definition 1.2.1. Let H be a Hilbert space. On B(H) we define the weak operator topologyτw as the coarsest topology for which all functions
fv,w:B(H)→C, A7→ hAv, wi, v, w∈ H,
are continuous. We define thestrong operator topology τsas the coarsest topol-ogy for which all maps
B(H)→ H, A7→Av, v∈ H,
are continuous. This topology is also called the topology of pointwise conver-gence.
Remark 1.2.2. (a) Since
fv,w(A)−fv,w(B) =h(A−B)v, wi ≤ k(A−B)vk · kwk
by the Cauchy–Schwarz Inequality, the functions fv,w are continuous onB(H) with respect to the strong operator topology. Therefore the weak operator topology is weaker (=coarser) than the strong one.
(b) In the weak operator topology all left and right multiplications λA:B(H)→B(H), X7→AX andρA:B(H)→B(H), X 7→XA are continuous. Indeed, forv, w∈ H, we have
fv,w(λA(X)) =hAXv, wi=fv,A∗w(X),
so thatfv,w◦λAis continuous, and this implies thatλAis continuous. Similarly, we obtainfv,w◦ρA=fAv,w, and hence the continuity ofρA.
Proposition 1.2.3. On the unitary groupU(H)the weak and the strong oper-ator topology coincide and turn it into a topological group.
We write U(H)s for the topological group (U(H), τs).
Proof. Forv∈ Handgi→gin U(H) in the weak operator topology, we have kgiv−gvk2=kgivk2+kgvk2−2 Rehgiv, gvi= 2kvk2−2 Rehgiv, gvi
→2kvk2−2 Rehgv, gvi= 0.
Therefore the orbit maps U(H)→ H, g7→gvare continuous with respect to the weak operator topology, so that the weak operator topology on U(H) is finer than the strong one. Since it is also coarser by Remark 1.2.2, both topologies coincide on U(H).
The continuity of the multiplication in U(H) is most easily verified in the strong operator topology, where it follows from the estimate
kgihiv−ghvk=kgi(hi−h)v+ (gi−g)hvk ≤ kgi(hi−h)vk+k(gi−g)hvk
=k(hi−h)vk+k(gi−g)hvk.
This expression tends to zero for gi → g and hi → h in the strong operator topology.
The continuity of the inversion follows in the weak topology from the conti-nuity of the functions
fv,w(g−1) =hg−1v, wi=hv, gwi=hgw, vi=fw,v(g) forv, w∈ Handg∈U(H).
Remark 1.2.4. (a) If dimH < ∞, then the norm topology and the strong operator topology coincide on B(H), hence in particular on U(H). In fact, choosing an orthonormal basis (e1, . . . , en) inH, we representA∈B(H) by the matrixA= (aij)∈Mn(C), whereaij =hAej, eii=fej,ei(A). IfEij ∈Mn(C) denote the matrix units, we then haveA=Pn
i,j=1aijEij, so that kAk ≤
n
X
i,j=1
|aij|kEijk=
n
X
i,j=1
|fej,ei(A)|kEijk,
which shows that convergence in the weak topology implies convergence in the norm topology.
(b) If dimH = ∞, then the strong operator topology on U(H) is strictly weaker than the norm topology. In fact, let (ei)i∈I be an orthonormal basis of H. ThenIis infinite, so that we may w.l.o.g. assume thatN⊆I. For eachnwe then define the unitary operatorgn ∈U(H) by gnei := (−1)δinei. For n6=m, we then have
kgn−gmk ≥ k(gn−gm)enk=k −2enk= 2, and
hgnv, wi − hv, wi=hgnv−v, wi=h−2hv, enien, wi=−2hv, enihen, wi →0 implies that limn→∞gn =1in the weak operator topology.
Definition 1.2.5. LetHbe a complex Hilbert space andGa topological group.
A continuous homomorphism
π: G→U(H)s
is called a (continuous) unitary representation ofG. We often denote unitary representations as pairs (π,H). In view of Proposition 1.2.3, the continuity of πis equivalent to the continuity of all therepresentative functions
πv,w:G→C, πv,w(g) :=hπ(g)v, wi.
A representation (π,H) is called norm continuous, if it is continuous with respect to the operator norm on U(H). Clearly, this condition is stronger
Here is a convenient criterion for the continuity of a unitary representation:
Lemma 1.2.6. A unitary representation (π,H) of the topological group G is continuous if and only if there exists a subset E⊆ Hfor which spanE is dense and the functions πv,w are continuous forv, w∈E.
Proof. The condition is clearly necessary because we may take E=H.
To see that it is also sufficient, we show that all functionsπv,w,v, w∈ H, are continuous. If F := spanE, then all functions πv,w, v, w ∈F, are continuous because the spaceC(G,C) of continuous functions onGis a vector space.
Let v, w ∈ H and vn → v, wn → w with vn, wn ∈ F. We claim that the sequenceπvn,wn converges uniformly toπv,w, which then implies its continuity.
In fact, for eachg∈Gwe have
|πvn,wn(g)−πv,w(g)|=|hπ(g)vn, wni − hπ(g)v, wi|
=|hπ(g)(vn−v), wni − hπ(g)v, w−wni|
≤ kπ(g)(vn−v)kkwnk+kπ(g)vkkw−wnk
=kvn−vkkwnk+kvkkw−wnk →0.
Example 1.2.7. If (ej)j∈J is an orthonormal basis ofH, thenE:={ej:j ∈J}
is a total subset. We associate toA∈B(H) the matrix (ajk)j,k∈J, defined by ajk:=hAek, eji,
so that
AX
k∈J
xkek =X
j∈J
X
k∈J
ajkxk ej.
Now Lemma 1.2.6 asserts that a unitary representation (π,H) of Gis con-tinuous if and only if all functions
πjk(g) :=hπ(g)ek, eji=πek,ej(g)
are continuous. These functions are the entries ofπ(g), considered as a (J×J )-matrix.
To deal with unitary group representations, we shall frequently have to deal with representations of more general structures, called involutive semigroups.
Definition 1.2.8. A pair (S,∗) of a semigroupS and an involutive antiauto-morphisms7→s∗is called an involutive semigroup. Then we have (st)∗=t∗s∗ fors, t∈S and (s∗)∗=s.
Example 1.2.9. (a) Any abelian semigroupSbecomes an involutive semigroup with respect tos∗ :=s.
(b) IfGis a group andg∗:=g−1, then (G,∗) is an involutive semigroup.
(c) An example of particular interest is the multiplicative semigroup S = (B(H),·) of bounded operators on a complex Hilbert spaceH(Example 1.1.12(a)).
Definition 1.2.10. (a) A representation (π,H) of the involutive semigroup (S,∗) is a homomorphismπ:S→B(H) of semigroups satisfyingπ(s∗) =π(s)∗ for eachs∈S.
(b) A representation (π,H) of (S,∗) is callednon-degenerate, ifπ(S)Hspans a dense subspace ofH. This is in particular the case if 1∈π(S).
(c) A representation (π,H) is called cyclicif there exists av ∈ Hfor which π(S)vspans a dense subspace ofH.
(d) A representation (π,H) is called irreducible if {0} and H are the only closedπ(S)-invariant subspaces ofH.
Example 1.2.11. If Gis a group with g∗ =g−1, then the representations of the involutive semigroup (G,∗) mapping1∈Gto1∈B(H), are precisely the unitary representations of G. All unitary representations of groups are non-degenerate sinceπ(1) =1.
Exercises for Section 1.2
Exercise 1.2.1. LetHbe a Hilbert space. Show that:
(1) The involution on B(H) is continuous with respect to the weak operator topology.
(2) On every bounded subset K ⊆B(H) the multiplication (A, B) 7→ AB is continuous with respect to the strong operator topology.
(3) On the unit sphere S := {x∈ H:kxk = 1} the norm topology coincides with the weak topology.
Exercise 1.2.2. LetHbe a Hilbert space and U(H)sits unitary group, endowed with the strong (=weak) operator topology. Show that the action map
σ: U(H)s× H → H, (g, v)7→gv
is continuous. Conclude that each continuous unitary representation (π,H) of a topological groupGdefines a continuous action ofGonHbyg.v:=π(g)v.
Exercise 1.2.3. Let (an)n∈N be a sequence of real numbers. Show that we obtain a continuous unitary representation of G= (R,+) on H=`2(N,C) by
π(t)x= (eita1x1, eita2x2, . . .).
Show further that, if the sequence (an) is unbounded, thenπis not norm con-tinuous. Is it norm continuous if the sequence (an) is bounded?
Exercise 1.2.4. Let (π,H) be a representation of an involutive semigroup (S,∗). Show that:
(a) (π,H) is non-degenerate if and only ifπ(S)v⊆ {0} impliesv= 0.
(b) Show that (π,H) is an orthogonal direct sum of a non-degenerate represen-tation and a zero represenrepresen-tation (ζ,K), i.e.,ζ(S) ={0}.
Exercise 1.2.5. Let (π,H) be a representation of the involutive semigroup (G, ηG), whereGis a group. Show that:
(a) (π,H) is non-degenerate if and only ifπ(1) =1.
(b) H=H0⊕ H1, where Hj = ker(η(1)−j1), is an orthogonal direct sum.
Exercise 1.2.6. Let (X, d) be a metric space andG:= Aut(X, d) be the group of automorphisms of (X, d), i.e., the group of bijective isometries. Show that the coarsest topology onGfor which all functions
fx:G→R, fx(g) :=d(g.x, x)
are continuous turns G into a topological group and that the action σ:G×X →X,(g, x)7→g.xis continuous.