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The Dixmier-Douady Invariant for Dummies Claude Schochet Wayne State University, [email protected]

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The Dixmier-Douady Invariant for Dummies Claude Schochet Wayne State University, Clsmath@Gmail.Com Wayne State University Mathematics Faculty Research Publications Mathematics 1-1-2009 The Dixmier-Douady Invariant for Dummies Claude Schochet Wayne State University, [email protected] Recommended Citation C. Shochet, The Dixmier-Douady invariant for dummies, Notices of the American Mathematical Society 56.7 (2009), 809-816. Available at: https://digitalcommons.wayne.edu/mathfrp/16 This Article is brought to you for free and open access by the Mathematics at DigitalCommons@WayneState. It has been accepted for inclusion in Mathematics Faculty Research Publications by an authorized administrator of DigitalCommons@WayneState. The Dixmier-Douady Invariant for Dummies Claude Schochet he Dixmier-Douady invariant is the pri- a map f to the classifying space BG and there is a mary tool in the classification of contin- pullback diagram ∗ uous trace C -algebras. These algebras G --------------------------------------------------------------------------------------------------------------------------------------! G have come to the fore in recent years ? ? ? ? Tbecause of their relationship to twisted y y K-theory and via twisted K-theory to branes, gerbes, T --------------------------------------------------------------------------------------------------------------------------------------! EG and string theory. ? ? This note sets forth the basic properties of ? ? y y the Dixmier-Douady invariant using only classical f homotopy and bundle theory. Algebraic topology X --------------------------------------------------------------------------------------------------------------------------------------! BG enters the scene at once since the algebras in where the right column is the universal principal question are algebras of sections of certain fibre G-bundle. bundles. Suppose further that F is some G-space. Then The results stated are all contained in the orig- following Steenrod [24] we may form the associated inal papers of Dixmier and Douady [5], Donovan fibre bundle and Karoubi [7], and Rosenberg [23]. Our treatment is novel in that it avoids the sheaf-theoretic tech- F -! T ×G F -! X niques of the original proofs and substitutes more with fibre F and structural group G. Pullbacks classical algebraic topology. Some of the proofs are commute with taking associated bundles, so there borrowed directly from the recent paper of Atiyah is a pullback diagram and Segal [1]. Those interested in more detail and especially in the connections with analysis should F --------------------------------------------------------------------------------------------------------------------------------------! F ? ? consult Rosenberg [23], the definitive work of Rae- ? ? burn and Williams [21], as well as the recent paper y y of Karoubi [13] and the book by Cuntz, Meyer, and T ×G F --------------------------------------------------------------------------------------------------------------------------------------! EG ×G F Rosenberg [4]. We briefly discuss twisted K-theory ? ? ? ? itself, mostly in order to direct the interested reader y y to some of the (exponentially-growing) literature f X ! BG: on the subject. -------------------------------------------------------------------------------------------------------------------------------------- It is a pleasure to acknowledge the assistance of Now suppose that M is some fixed C∗-algebra, Alan Carey, Dan Isaksen, Max Karoubi, N. C. Phillips, soon to be either the matrix ring Mn = Mn(C) and Jonathan Rosenberg in the preparation of this for some n < 1 or the compact operators K on paper. some separable Hilbert space H . Take G = U(M), the group of unitaries of the C∗-algebra. (If M Fibre Bundles is not unital then we modify by first adjoining Suppose that G is a topological group and G ! a unit canonically to form the unital algebra M+ T ! X is a principal G-bundle over the compact and then define U(M) to be the kernel of the nat- space X. Then up to equivalence it is classified by ural homomorphism U(M+) ! U(M+=M) S1.) Then U(M) acts naturally on M by conjugation; ad Claude Schochet is professor of mathematics at Wayne denote M with this action as M . The center State University, Detroit, MI. His email address is claude@ ZU(M) of U(M) acts trivially, and so the action math.wayne.edu. descends to an action of the projective unitary August 2009 Notices of the AMS 809 group PU(M) = U(M)/ZU(M) on M, denoted Mad . classifying spaces. The one that concerns us is Note that if M is simple then its center is just C and the tensor product operation. Fix some unitary ZU(M) S1 so that PU(M) is just the quotient isomorphism of vector spaces group (U(M))/S1. r s rs Having fixed M, let C ⊗ C C : ζ : PU(M) -! T ! X (This isomorphism is unique up to homotopy, since be a principal PU(M)-bundle over a compact space the various unitary groups are connected.) Let X. Form the associated fibre bundle Un = U(Mn(C)). This determines a homomorphism ad ad p ⊗ Pζ : M -! T ×PU(M) M -! X: Ur × Us -! Urs This fibre bundle always has non-trivial sections. and the associated map on classifying spaces Define Aζ to be the space of sections: ⊗ ad BUr × BUs -! BUrs Aζ = Ð (Pζ) = fs : X -! T ×PU(M) M j ps = 1g: This is a C∗-algebra with pointwise operations that given by the composite are well defined because we are using the adjoint B(⊗) action. It is unital if M is unital. If M = Mn or BUr × BUs B(Ur × Us ) -! BUrs : M = K then this is a continuous trace C∗-algebra. Let [X; Y ] denote homotopy classes of maps and If X is locally compact but not compact then Aζ is still defined by using sections that vanish at recall that if X is compact and connected then infinity and it is not unital. isomorphism classes of complex n-plane vector Note that if Pζ is a trivial fibre bundle then bundles over X correspond to elements of [X; BUn]. sections correspond to functions X ! M and hence Then this construction induces an operation ⊗ Aζ C(X) ⊗ M; [X; BUr ] × [X; BUs ] -! [X; BUrs ], ∗ where C(X) denotes the C -algebra of continuous which does indeed correspond to the tensor prod- complex-valued functions on X. (If X is only locally uct operation on bundles. Precisely, if E ! X and compact then we use C to denote continuous 1 o E ! X are represented by f and f respectively, functions vanishing at infinity.) 2 1 2 then the tensor product bundle E ⊗ E ! X is Continuous trace C∗-algebras may be defined 1 2 represented by f ⊗ f . (This holds at once for intrinsically, of course. Here is one approach. If A 1 2 compact connected spaces. If X is not connected is a (complex) C∗-algebra, then let Aˆ denote the then one checks this on each component.) set of unitary equivalence classes of irreducible ∗-representations of A with the Fell topology (cf. The inclusion [21]). Ur Ur × f1g ! Ur × Us ! Urs Definition. Let X be a second countable locally is denoted compact Hausdorff space. A continuous trace C∗- algebra with Aˆ = X is a C∗-algebra A with Aˆ = X αrs : Ur ! Urs : such that the set 1 The center of Uk is the group S regarded as ma- fx 2 A j the map π ! tr(π(a)π(a)∗) trices of the form zI, where z is a complex number is finite and continuous on Aˆg of norm 1. The quotient group PUk is the projec- 1 tive unitary group. The fibration S ! Uk ! PUk is dense in A. induces the sequence From the definition it is easy to see that com- k ∗ 0 ! Z -! Z ! Z=k ! 0 mutative C -algebras Co(X) as well as stable ∗ K commutative C -algebras Co(X; Mn) and Co(X; ) on fundamental groups, and, in particular, are continuous trace. In fact every continuous trace π (PU ) Z=k. algebra arises as a bundle of sections of the type 1 k There is a natural induced map and commuting we have been discussing. diagram Products S1 × S1 --------------------------------------------------------------------------------------------------------------------------------------! S1 Vector spaces come equipped with natural direct ? ? ? ? sum and tensor product operations, and these pass y y over to vector bundles. Thus if E1 ! X and E2 ! X are complex vector bundles of dimension r and s Ur × Us --------------------------------------------------------------------------------------------------------------------------------------! Urs ? ? respectively then we may form bundles E1 ⊕E2 ! X ? ? y y of dimension r + s and E1 ⊗ E2 ! X of dimension rs. There are two corresponding operations on PUr × PUs --------------------------------------------------------------------------------------------------------------------------------------! PUrs 810 Notices of the AMS Volume 56, Number 7 and this induces a tensor product operation and a Proposition ([8]). On the category of compact commuting diagram spaces, Cechˇ cohomology is representable. That is, BUr × BUs --------------------------------------------------------------------------------------------------------------------------------------! BUrs there is a natural isomorphism ? ? n ? ? Hˇ (X; Z) [X; K(Z; n)]. y y Proof. The natural isomorphism is well known BPU × BPU --------------------------------------------------------------------------------------------------------------------------------------! BPU : r s rs for X a finite complex, by the Eilenberg-Steenrod It is easy to see that uniqueness theorem. Suppose that X is a compact π (BPU ) π (PU ) Z=k space. Then write X = lim X for some inverse 2 k 1 k ----------------------------------- j system of finite complexes. (See [8], Chapters IX, X, and that the natural map αrs : PUr ! PUrs induces a commuting diagram and XI for open covers, nerves, and inverse limits.) Continuity of Cechˇ theory implies that (αrs )∗ π2(BPr ) --------------------------------------------------------------------------------------------------------------------------------------
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