1. Consider the operators ut , t ∈ R, on L2 (R) defined by (ut f )(s) = f (s − t). Show that
ut → ut0 in the strong operator topology as t → t0 , but kus − ut k = 2 for all s 6= t.
2. Consider the algebra L∞ (0, 1) acting on L2 (0, 1) by multiplication. Show that L∞ (0, 1)0 =
L (0, 1). Therefore L∞ (0, 1) ⊂ B(L2 (0, 1)) is a von Neumann algebra.
∞
3. Recall that the tensor product of vector spaces U and V is a vector space U ⊗ V
together with a bilinear map U × V → U ⊗ V , denoted by (u, v) 7→ u ⊗ v, with the folllowing
universal property: if B 7→ U × V → W is a bilinear map then there exists a unique linear
map B̄ : U ⊗ V → W such that B̄(u ⊗ v) = B(u, v), that is, the diagram
6U
mmm
m
m
mm
mmm
mmm
⊗V
B̄
U ×V R
RRR
RRR
RRR
RRR
B
RR( W
commutes.
Show that
(i) if U ⊗ V exists then it is unique up to a canonical isomorphism;
(ii) the vector spaces U ⊗ (⊕i Vi ) and ⊕i (U ⊗ Vi ) are canonically isomorphic.
4. Show that
(i) if {ei }i is an orthonormal basis in a Hilbert space H1 and {fj }j is an orthonormal basis
in a Hilbert space H2 , then {ei ⊗ fj }i,j is an orthonormal basis in H1 ⊗ H2 ;
(ii) if (X, µ) and (Y, ν) are measure spaces then L2 (X, µ) ⊗ L2 (Y, ν) ∼
= L2 (X × Y, µ × ν).
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