By Matilde Marcolli, Deepak Parashar

This ebook is aimed toward proposing varied equipment and views within the thought of Quantum teams, bridging among the algebraic, illustration theoretic, analytic, and differential-geometric ways. It additionally covers fresh advancements in Noncommutative Geometry, that have shut relatives to quantization and quantum team symmetries. the quantity collects surveys by way of specialists which originate from an acitvity on the Max-Planck-Institute for arithmetic in Bonn.

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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).