By Dana P. Williams

The idea of crossed items is very wealthy and fascinating. There are functions not just to operator algebras, yet to topics as assorted as noncommutative geometry and mathematical physics. This publication presents a close advent to this mammoth topic appropriate for graduate scholars and others whose study has touch with crossed product $C^*$-algebras. as well as supplying the fundamental definitions and effects, the focus of this ebook is the wonderful excellent constitution of crossed items as printed by means of the research of brought about representations through the Green-Mackey-Rieffel laptop. particularly, there's an in-depth research of the imprimitivity theorems on which Rieffel's concept of triggered representations and Morita equivalence of $C^*$-algebras are dependent. there's additionally a close therapy of the generalized Effros-Hahn conjecture and its facts as a result of Gootman, Rosenberg, and Sauvageot. This publication is intended to be self-contained and obtainable to any graduate pupil popping out of a primary direction on operator algebras. There are appendices that take care of ancillary topics, which whereas no longer critical to the topic, are however an important for an entire knowing of the fabric. a few of the appendices can be of autonomous curiosity. To view one other e-book by way of this writer, please stopover at Morita Equivalence and Continuous-Trace $C^*$-Algebras

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**Example text**

First, the fact that Eλ and Eρ are idempotents follows immediately from the Fubini-Tonelli theorem together with translation invariance of the Haar measure μ, whilst commutation of Eλ and Eρ follows from the Fubini-Tonelli theorem together with the commutation of left and right actions on H1 and on H2 . Moreover, by construction, Eλ and Eρ act as the identity on LLA (H1 , H2 ) and LR A (H1 , H2 ), respectively, so that im(Eλ ) ⊇ LLA (H1 , H2 ), im(Eρ ) ⊇ LR A (H1 , H2 ). Now, let T ∈ L1A (H1 , H2 ).

13) (mult ◦ιn )(Bimod(A, n)) = SymS (Z≥0 ). Proof. First, since a unitary equivalence of real A-bimodules of KO-dimension n mod 8 is, in particular, a unitary equivalence of odd A-bimodules, the map ιn is well deﬁned. Next, let (H, J) and (H , J ) be real A-bimodules of KO-dimension n mod 8, and suppose that H and H are unitarily equivalent as bimodules; let U ∈ ULR A (H , H). Now, if m is the multiplicity matrix of H, then H and Hm are unitarily ∗ equivalent, so let V ∈ ULR and V U J U ∗ V ∗ are both real A (H, Hm ).

5. Real spectral triples of even KO-dimension. We now turn to real spectral triples of even KO-dimension. Because of the considerable qualitative diﬀerences between the two cases, we consider separately the case of KO-dimension 0 or 4 mod 8 and KO-dimension 2 or 6 mod 8. In what follows, (H, γ, J) is a ﬁxed real A-bimodule of even KO-dimension n mod 8 with multiplicity matrices (meven , modd ); we denote by L1A (Heven , Hodd ; J) the subspace of L1A (Heven , Hodd ) consisting of δ such that 0 Δ∗ Δ 0 ∈ D0 (A, H; γ, J).