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By Samuel Moy

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The term Haar measure is reserved for left-invariant measures. 56. Our preference for left over right Haar measures is strictly convention. If µ is a left Haar measure, then ν(E) := µ(E −1 ) is a right Haar measure. 67 on page 20. The fundamental result is the following. 57. Every locally compact group G has a Haar measure which is unique up to a strictly positive scalar. 57 need not concern us here. 1. For the moment, be aware that the precise formulation of the regularity conditions (given by the supremum and infimum conditions) varies a bit from reference to reference (if the group is not second countable).

But u∗s (x) − u∗t (x) 2 A = u∗s (x) − u∗t (x) , u∗s (x) − u∗t (x) =2 x 2 A − us u∗t (x) , x But s → t, implies us u∗t (x) → x by assumption. A A − x , us u∗t (x) A . 100. 10], we exhibit a sequence S := { Tn } ⊂ B(H) such that 0 belongs to the ∗-strong closure of S but not to the strict closure. Therefore the strict and ∗-strong topologies are not the same. Notice too that the principle of uniform boundedness implies that a strongly convergent sequence is bounded. 98 on the preceding page. Therefore there must be a net in S converging to 0 ∗-strongly, but no sequence in S can converge to 0 ∗strongly.

We are going to study dynamical systems and their crossed products via classes of representations called covariant representations. We define dynamical systems, their crossed products and covariant representations in the first two sections. Just as characters are in one-to-one correspondence with complex homomorphisms (that is, one-dimensional representations) of the L1 -algebra of a locally compact abelian group, covariant representations of a dynamical system are in one-to-one correspondence with representations of the associated crossed product.

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An introduction to the theory of field extensions by Samuel Moy


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