By Karl H. Hofmann

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**Example text**

Define a representation σ : L∞ (T) → B( 2 (Z, L2 (T))) by (σ(f )η)(n) = f η(n), Then, for f, g ∈ L∞ (T), η ∈ f ∈ L∞ (T), η ∈ 2 2 (Z, L2 (T)), n ∈ Z. 7) showing that σ(f ) commutes with π(g). Similarly, for k ∈ Z, (σ(f )λk η)(n) = f (λk η)(n) = f η(n − k) = λk (σ(f )η)(n), and σ(f ) also commutes with each λk . Thus σ(f ) ∈ (L∞ (T) f ∈ L∞ (T). Now define unitaries un ∈ B(L2 (T)) by (un g)(e2πit ) = g(e2πi(t+nθ) ), g ∈ L2 (T), n ∈ Z. 10) for f ∈ L∞ (T), g ∈ L2 (T). This shows that un mf u∗n = mαn (f ) for f ∈ L∞ (T).

We then define an action of Z on L (T) by αn (f )(e2πit ) = f (e2πi(t+nθ) ), f ∈ L∞ (T), t ∈ [0, 1], n ∈ Z. 3) The crossed product L∞ (T) α Z is then generated by the operators {π(f ) : f ∈ L∞ (T)} and {λn : n ∈ Z}. This is the weak closure of the C ∗ -algebra generated by these operators, known as the irrational rotation C ∗ -algebra since Z acts by irrational rotation. Consider the vector ξ ∈ 2 (Z, L2 (T)) whose only non-zero 30 CHAPTER 3. FINITE VON NEUMANN ALGEBRAS component is ξ(0) = 1. Then let τ be the vector state ·ξ, ξ .

If b ∈ F , then hbh−1 ∈ / F and it such that k −1 hk ∈ / B. On the other hand, if b ∈ B ∩ H, then clearly cannot lie in H, so hbh−1 ∈ hbh−1 = b. These two cases show that (C3) is implied by (C4), completing the proof. In the following example we show that groups satisfying (C1) and (C3) exist, leading to an example of a semi-regular masa. 5. 3, we let F be an infinite field and we define a 0 G= x 1 : a ∈ F ×, x ∈ F , 1 x 0 1 H= : x∈F . 18) We regard these as lying in GL2 (F ), the group of invertible 2 × 2 matrices over F , whose centre Z is the set of non-zero multiples of the identity.