THE EXTENSION CLASS AND KMS STATES FOR CUNTZ–PIMSNER ALGEBRAS OF SOME BI-HILBERTIAN BIMODULES ADAM RENNIE, DAVID ROBERTSON, AND AIDAN SIMS Abstract. Forbi-HilbertianA-bimodules, inthe senseofKajiwara–Pinzari–Watatani,we construct a Kasparov module representing the extension class defining the Cuntz–Pimsner algebra. The construction utilises a singular expectation which is defined using the C∗- module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz–Pimsner algebras. 6 1 0 2 r a 1. Introduction M The Cuntz–Pimsner algebras introduced in [35] have attracted enormous attention over 2 the last fifteen years (see, for example, [1, 6, 7, 8, 9, 13, 18, 19, 22, 24, 26, 28, 32, 37]). ] They are at once quite tractable and very general, including models for crossed products T and Cuntz–Krieger algebras [35], graph C∗-algebras [12], topological-graph C∗-algebras [21], K Exel crossed products [4], C∗-algebras of self-similar actions [33] and many others. . h Particularly important in the theory of Cuntz–Pimsner algebras is the natural Toeplitz at extension 0 End0(F ) T O 0 of a Cuntz–Pimsner algebra by the compact m → A E → E → E → endomorphisms of the associated Fock module. For example, Pimsner uses this extension in [ [35] to calculate the K-theory of a Cuntz–Pimsner algebra using that End0(F ) is Morita 2 equivalent to A and T is KK-equivalent to A. It follows that the class oAf thEis extension E v is important in K-theory calculations, and a concrete Kasparov module representing this 3 6 class could be useful, for example, in exhibiting Poincar´e duality for appropriate classes of 3 Cuntz–Pimsner algebras. 5 When E is an imprimitivity bimodule, this is relatively straightforward (see Section 3.1) 0 . because the Fock representation of T is the compression of a natural representation of O 1 E E 0 on a 2-sided Fock module. But for the general situation, there is no such 2-sided module. 5 Pimsner sidesteps this issue in [35] by replacing the coefficient algebra A with the direct 1 limit A of the algebras of compact endomorphisms on tensor powers of E, and E with : ∞ v the direct limit E of the modules of compact endomorphisms from E⊗n to E⊗n+1. This is i ∞ X an excellent tool for computing the K-theory of O : the module E (rather than E itself) E ∞ r induces the Pimsner–Voiculescu sequence in K-theory, and the Cuntz–Pimsner algebra of a E is isomorphic to that of E. But at the level of KK-theory, replacing E with E changes ∞ ∞ things dramatically. The Toeplitz extension associated to E corresponds to an element of ∞ KK1(O ,A ), rather than of KK1(O ,A), and the two are quite different: for example, E ∞ E if E is the 2-dimensional Hilbert space, then A = C, whereas A = M (C), the type-2∞ ∞ 2∞ UHF algebra. 2010 Mathematics Subject Classification. 19K35, 46L08,46L30. Key words and phrases. Kasparov module; extension; Cuntz–Pimsner algebra;KMS state. This research was supported by the Australian Research Council. 1 2 ADAMRENNIE,DAVIDROBERTSON,ANDAIDANSIMS In this paper we consider the situation where E is a finitely generated bi-Hilbertian bi- module, in the sense of Kajiwara–Pinzari–Watatani [18], over a unital C∗-algebra. Our main result is a construction of a Kasparov-module representative of the class in KK1(O ,A) E corresponding to the extension of O by End0(F ). We assume our modules E are both E A E full and injective. This situation is quite common, and we present a range of examples; but much that we do could be extended to more general finite-index bi-Hilbertian bimodules, [18]. After introducing some basic structural features of the modules we consider in Section 2, we give a range of examples. We then examine the important special case of self-Morita equivalence bimodules (SMEBs), which include crossed products by Z. This case was first calculated by Pimsner [35] in order to show that A and T are KK-equivalent. We present E the details here for completeness. For SMEBs we can produce an unbounded representative of the extension 0 End0(F ) T O 0 → A E → E → E → defining O . Here E is our correspondence, F the (positive) Fock space, and T , O are E E E E the Toeplitz–Pimsner and Cuntz–Pimsner algebras, respectively. Having an unbounded representative can simplify the task of computing Kasparov pro- ducts. Since products with the class of this extension define boundary maps in K-theory and K-homology exact sequences, this representative is a useful aid to computing K-theory via the Pimsner–Voiculescu exact sequence. An application of this technique to the quantum Hall effect appears in [3]. For the general case of (finitely generated) bi-Hilbertian bimodules, we do not obtain an unbounded representative, but the construction of the right A-module underlying the Kasparov module is novel. Using the bimodule structure, we construct a one-parameter family Φ : T A, (s) > 1, of positive A-bilinear maps. Provided the residue at s = 1 s E exists, we obtain→an exℜpectation Φ = res Φ : T A, which vanishes on the covariance ∞ s=1 s E ideal, and so descends to O . We use Φ to construct→an A-valued inner product on O , and E ∞ E thereby obtain the underlying C∗-module in our O –A-Kasparov module representing the E extension class. We provide a criterion for establishing the existence of the desired residue in Proposition 3.5. We show that this criterion is readily checkable in some key examples; in particular, we show in Example 3.8 that the residue exists when E is the bimodule associated to a finite primitive directed graph. The bimodule structure and Jones–Watatani index are essential ingredients in the con- struction of Φ . The (right) Jones–Watatani index also provides a natural and interesting ∞ one-parameter family of quasi-free automorphisms of O , andwe show that thereis a natural E family of KMS states on O parameterised by the states on A which are invariant for the E dynamics encoded by E. This construction combines ideas from [25] and [5]. There are also corresponding dynamics arising from the left Jones–Watatani index, and the product of the left and right indices. The corresponding collections of KMS states would also be interesting, but we do not address them here. The key point is that many important Cuntz–Pimsner algebras arise from bi-Hilbertian modules, and this extra structure gives rise to new tools that are worthy of study. Acknowledgements. This work has profited from discussions with Bram Mesland and Magnus Goffeng. The authors also wish to thank the anonymous referee for several sugges- tions which have greatly improved the exposition. THE EXTENSION CLASS AND KMS STATES 3 2. A class of bimodules Throughout this paper, A will denote a separable, unital, nuclear C∗-algebra. Given a rightHilbertA-moduleE (writtenE whenwewanttoremember thecoefficient algebra), we A denote the C∗-algebra of adjointable operators by End (E), the compact endomorphisms by A End0(E) and the finite-rank endomorphisms by End00(E). The finite-rank endomorphisms A A are generated by rank one operators Θ with e, f E. e,f ∈ Definition 2.1. Let A be a unital C∗-algebra. Following [18], a bi-Hilbertian A-bimodule is a full right C∗-A-module with inner product ( ) which is also a full left Hilbert A-module A · | · with inner product ( ) such that the left action of A is adjointable with respect to ( ) A A · | · · | · and the right action of A is adjointable with respect to ( ). A · | · If E is a bi-Hilbertian A-bimodule, then there are two Banach-space norms on E, arising from the two inner-products. The following straightforward lemma shows that these norms are automatically equivalent. Lemma 2.2. Let E be a bi-Hilbertian A-bimodule. Then there are constants c,C R such ∈ that (e e) c (e e) and (e e) C (e e) for all e E. A A A A k | k ≤ k | k k | k ≤ k | k ∈ Proof. By symmetry it suffices to find c. Suppose that no such c exists. Then there is a sequence e E such that (e e ) > n (e e ) . By normalising, we may assume n n n A A n n ∈ k | k k | k that each (e e ) = 1, and hence each (e e ) < 1. So e 0 in E, and then k n | n Ak kA n | n k n n → (cid:3) continuity of ( ) forces (0 0) = 1, contradicting the inner-product axioms. A A · | · k | k Throughout the paper, if we say that A is a finitely generated projective bi-Hilbertian A-bimodule, we mean that it is finitely generated and projective both as a left and as a right A-module. The next lemma characterises when a right A-module has a left inner product for a second algebra. It provides a noncommutative analogue of ‘the trace over the fibres’ for endomor- phisms of vector bundles. For us, a frame for a right-Hilbert module EA is a sequence (ei)i∈N of elements such that the series Θ converges strictly to Id ; note that this would be called a countable right ei,ei E basis in the terminology of [18], or a standard normalised tight frame in the terminology P of [14]. As discussed in [18, Section 1], every countably generated Hilbert module E over a σ-unital C∗-algebra A admits a frame (in our sense), and it admits a finite frame if and only if End0(E) = End (E). As discussed in the remark following [18, Proposition 1.2], if (e ) is A A i a frame for E, then the net Θ (f) indexed by finite subsets F of e converges to i∈F ei,ei { i} f for all f E. ∈ P A less general version of the following basic lemma appears in [29, Lemma 3.23]. Lemma 2.3 (cf. [18, Lemma 2.6]). Let E be a countably generated right-Hilbert A-module, A and let B End (E) be a C∗-subalgebra. A ⊂ (1) Suppose that ( ) is a left B-valued inner product for which the right action of A B · | · is adjointable. Then there is a B-bilinear faithful positive map Φ : End00(E) B A → such that Φ(Θ ) := (e f) for all e,f E. For any frame (e ) for E, we have e,f B i | ∈ Φ(T) = (Te e ) for all T End00(E). iB i | i ∈ A P 4 ADAMRENNIE,DAVIDROBERTSON,ANDAIDANSIMS (2) Suppose thatΦ : End0(E) B isa B-bilinearfaithful positivemap. Then (e f) := A → B | Φ(Θ ) defines a left B-valued inner product on E for which the right action of A is e,f adjointable. Proof. (1) Choose a frame (e ) for E. Given a rank-one operator Θ , using the frame i e,f property at the last equality, we calculate: (Θ e e ) = (e(f e ) e ) = (e e (e f) ) B e,f i i B i A i B i i A | | | | | i i i X X X = (e Θ f) = (e f). B | ei,ei B | i X It follows that there is a well-defined linear map Φ : End00(E) B satisfying Φ(Θ ) = A → e,f (e f) as claimed. The remaining properties of Φ follow from straightforward calculations. B | For example, Φ(b Θ b ) = Φ(Θ ) = (b e b∗f) = b (e f)b = b Φ(Θ )b , 1 e,f 2 b1e,b∗2f B 1 | 2 1B | 2 1 e,f 2 so Φ is B-bilinear. Positivity and faithfulness follow from the corresponding properties of the inner product. (2) Given Φ : End0(E) B, we define A → (e f) := Φ(Θ ). B e,f | Since (e,f) Θ is a left End0(E)-valued inner-product on E, and since Φ is faithful 7→ e,f A and B-linear, it is routine to check that ( ) is positive definite. Since Φ is positive, it B is -preserving, and so (e f) = (f e)∗. ·W| r·ite ϕ for the homomorphism B End (E) B B A ∗ | | → that implements the left action. Then B-linearity of Φ gives b (e f) = bΦ(Θ ) = Φ(ϕ(b)Θ ) = Φ(Θ ) = (b e f). B e,f e,f b·e,f B | · | So Φ is a left B-valued inner product. For adjointability of the right A-action, observe that (e a f) = Φ(Θ ) = Φ(Θ ) = (e f a∗). (cid:3) B e·a,f e,f·a∗ B · | | · Remark 2.4. Unlike the Hilbert space case, the preceding result does not give any automatic cyclicity properties for the map Φ (which we might otherwise be tempted to regard as an operator-valued trace): for e,f E and U End (E) unitary, we have A ∈ ∈ Φ(Θ U) = (e U∗f) and Φ(UΘ ) = (Ue f). e,f B e,f B | | The adjoint U∗ in the first expression is the adjoint with respect to the inner-product ( ) , A · | · which is the inverse of U. However, it is not clear that U−1 is an adjoint for U with respect to ( ), even assuming that U is adjointable for ( ). B B · | · · | · Remark 2.5. Consider the (very) special case where A is commutative, E is a symmetric A-bimodule in the sense that a e = e a for all e E, and (e f) = (f e) . Then the A A · · ∈ | | operator-valued weight associated to ( ) is a trace: given Θ and Θ , A e,f g,h · | · Φ(Θ Θ ) = Φ(Θ ) = (e(f g) h) e,f g,h e(f|g)A,h A | A | and Φ(Θ Θ ) = (g(h e) f). g,h e,f A A | | THE EXTENSION CLASS AND KMS STATES 5 The following computation shows that these are equal. (e(f g) h) = (h e(f g) ) = (h e) (f g) = (f(e h) g) A A A A A A A A | | | | | | | | = (g f(e h) ) = (g(h e) f). A A A A | | | | Thus for vector bundles we recover the trace over the fibres of endomorphisms. Remark 2.6. If T End0(E) commutes with all b B then Φ(T) Z(B), because ∈ A ∈ ∈ bΦ(T) = Φ(bT) = Φ(Tb) = Φ(T)b. 2.1. Examples. We devote the remainder of this section to showing that many familiar and important classes of correspondences give rise to bi-Hilbertian bimodules of the type we consider. 2.1.1. Self-Morita equivalence bimodules (SMEBs). The following examples all share the important property that both the left and right endomorphism algebras are isomorphic to the coefficient algebra (or its opposite). This will turn out to be an important hypothesis, and also covers many important examples. Definition 2.7. Let A be a C∗-algebra. A self-Morita equivalence bimodule (SMEB) over A is a bi-Hilbertian A-bimodule E whose inner products are both full and satisfy the im- primitivity condition (e f)g = e(f g) , for all e, f, g E. A A | | ∈ Recall from [36, Proposition 3.8] that if E is a self-Morita equivalence A-bimodule, then A A = End0(E). ∼ A Example 2.8 (Crossed products by Z). Suppose that A is unital and nuclear, and let α : A A be an automorphism. Then the C∗-correspondence A with the usual right module α A → structure, left action of A determined by α and left inner product (a b) = α−1(ab∗) is a A | SMEB. The imprimitivity condition follows from the calculation a (b c) = ab∗c = α(α−1(ab∗))c = (a b) c. A A · | | · Example 2.9 (Line bundles). Suppose that A is unital and commutative, so that A = C(X) ∼ forsomesecond-countable compact Hausdorffspace X. Given acomplex linebundle L X, → we obtain a SMEB over A by setting E = Γ(L), the continuous sections of L. The left and right actions are by pointwise multiplication, and any Hermitian form , on L determines h· ·i inner products by (e f)(x) := e(x),f(x) =: (f e) . A A | h i | The next result shows that for SMEBs, the map Φ of Lemma 2.3 is an expectation. Lemma 2.10. Suppose that E is a SMEB over a unital C∗-algebra A. The map Φ : End0(E) A of Lemma 2.3(1) satisfies Φ(Id ) = 1 . A → E A Proof. Choose a frame (e ) for E. Since E is a SMEB, [36, Proposition 3.8] says that the i map Θ (x y) determines an isomorphism ψ : End0(E) A. In particular, ψ is x,y 7→ A | A → unital, and so 1 = ψ(Id ) = ψ Θ = (e e ) = Φ(Id ). (cid:3) A E ei,ei A i | i E i i (cid:0)X (cid:1) X Conversely, Corollary4.14of [18] shows thata bi-Hilbertianbimodule E satisfies Φ(Id ) = E 1 if and only E can be given a left inner product which makes E into a SMEB. A 6 ADAMRENNIE,DAVIDROBERTSON,ANDAIDANSIMS 2.1.2. Crossed products by injective endomorphisms. Let A be a unital C∗-algebra and sup- pose that α : A A is an injective unital -endomorphism. Assume there exists a faithful → ∗ conditional expectation Φ : A α(A). Then L := α−1 Φ is a transfer operator [11, → ◦ Definition 2.1]; that is, L : A A is a positive linear map satisfying → L(α(a)b) = aL(b) for all a,b A. ∈ There is a bi-Hilbertian A-bimodule associated to the triple (A,α,L) as follows: A is a pre-Hilbert A-bimodule with a e b := aeα(b) · · and (e f) := L(e∗f) A | for a,b,e,f A. Denote by E the completion of A for the norm e 2 = (e e) . By A ∈ k k k | k faithfulness of Φ, there is a left inner-product (e f) = ef∗ A | which is left A-linear and for which the right action of A on E is adjointable. The associated Cuntz–Pimsner algebra satisfies O = A⋊ N E α,L where A⋊ N is as defined by Exel [11]. α,L 2.1.3. Vector bundles. If E X is a complex vector bundle over a compact Hausdorff → space, then the C(X)-module Γ(E) of all continuous sections under pointwise multiplication is finitely generated and projective for any nondegenerate C(X)-valued inner products (left and right). If we alter the left action by composing with an automorphism, we also need to alter the left inner product as in Example 2.8. If E is rank one then we are back in the SMEB case of Example 2.9. 2.1.4. Topological graphs. Atopologicalgraphis a quadruple G = (G0,G1,r,s)where G0,G1 are locally compact Hausdorff spaces, r : G1 G0 is a continuous map and s : G1 G0 → → is a local homeomorphism. For simplicity, we will assume that r and s are surjective. Given a topological graph G, Katsura [21] associates a right Hilbert module as follows. Let A = C (G0). Then, similarly to Section 2.1.5, C (G ) is a right pre-Hilbert A-module with 0 c 1 left and right actions (a e b)(g) := a(r(g))e(g)b(s(g)), e C (G1), a,b A, g G1 c · · ∈ ∈ ∈ and inner product (e f) (v) = e(g)f(g), e,f C (G1), v G0. A c | ∈ ∈ s(g)=v X (Since s is a local homeomorphism, g vG1 : e(g) = 0 is finite for e C (G1), so this c { ∈ 6 } ∈ formula for the inner-product makes sense.) We write E for the completion of C (G1) in the c norm determined by the inner-product, and E is a right Hilbert A-module. To impose a left Hilbert module structure onE, we restrict attention to topological graphs where r is also a local homeomorphism, and define (e f)(v) = e(g)f(g), e,f C (G1), v G0. A c | ∈ ∈ r(g)=v X THE EXTENSION CLASS AND KMS STATES 7 For the remainder of this section, suppose that G0 and G1 are compact. The following lemma and its proof are due to Mitch Hawkins, [16]. Lemma 2.11. Suppose that r, s : G1 G0 are local homeomorphisms. For each n N, the → ∈ sets v G0 : G1v = n and w G0 : vG1 = n are compact open. { ∈ | | } { ∈ | | } Proof. We show that v G0 : G1v = n is compact open; symmetry does the rest. It { ∈ | | } suffices to show that v G0 : G1v n is both closed and open. { ∈ | | ≥ } First suppose that v satisfies G1v n. Fix distinct e ,...,e G1v. Since G1 is Haus- 1 n | | ≥ ∈ dorff, we can pick disjoint open neighbourhoods U of e . Since s is a local homeomorphism, i i we can shrink the U so that s(U ) = s(U ) for all i,j n and so that s is a homeomor- i i j ≤ |Ui phism for each i. Since s is a local homeomorphism, it is an open map, and so V = s(U ) 1 is an open neighbourhood of v. For each v′ V each U v′ is a singleton, and the U are i i ∈ mutually disjoint, so G1v′ n. Hence V v G0 : G1v n , and we deduce that the | | ≥ ⊆ { ∈ | | ≥ } latter is open. We now show that it is also closed. Suppose that v is a sequence in G0 converging to m v, and suppose that each G1v n. For each m, choose distinct elements e ,...,e m m,1 m,n | | ≥ of G1v . Since G1 is compact, by passing to a subsequence we may assume that each m sequence e converges to some e G1. By continuity of s, we have s(e ) = v for each m,i i i ∈ i, so it suffices to show that i = j implies e = e . For this, fix a neighbourhood U of e i j i 6 6 on which s is a homeomorphism. Since e e , the e eventually belong to U. Since m,i i m,i → each s(e ) = v = s(e ) and each e = e , we see that e U for large m. Since m,j j m,i m,j m,i m,j i e e , we deduce that e U, and in p6articular e = e . 6∈ (cid:3) m,j j j j i → 6∈ 6 Corollary 2.12. For m,n N, let ∈ G1 := e G1 : r(e)G1 = m and G1s(e) = n . m,n { ∈ | | | | } Then the G1 are compact open sets, as are s(G1 ) and r(G1 ). m,n m,n m,n Proof. We have G1 = s−1( v : G1v = n ) r−1( w : wG1 = m ). Lemma 2.11 and m,n { | | } ∩ { | | } continuity of s and r imply that G1 is clopen; since G1 is compact, each G1 then also m,n m,n compact. Since r,s are local homeomorphisms, they are open maps, so r(G1 ) and s(G1 ) m,n m,n are open. They are compact as they are continuous images of the compact set G1 . (cid:3) m,n Since r,s are local homeomorphisms, each edge e has a neighbourhood U on which both e s and r are homeomorphisms. By the preceding corollary, we may assume that each U e G1 . The U cover G1, so by compactness, there is a finite open cover U such tha⊆t |r(e)G1|,|G1s(e)| e each U U is contained in some G1 . Choose a partition of unity on G1 subordinate ∈ m(U),n(U) to U; say f : U U . So 0 f 1 and f C (U) for each U U, and f (e) = 1 { U ∈ } ≤ U ≤ U ∈ 0 ∈ U∈U U for all e G1. ∈ P Lemma 2.13. For each U U, define h C(G1) by h (e) := f (e). The collection U U U h : U U is a frame for∈both the left an∈d the right inner-product on C(G1). We have U { ∈ } p Φ(Id )(v) = vG1 for all v G0. E | | ∈ Proof. The situation is completely symmetrical in r and s, so we just have to show that the f form a frame for the right inner-product. For this, we fix g C(G1) and e G1 and U ∈ ∈ calculate (2.1) (θ g)(e) = h (e)(h g) (s(e)) = f (e) f (e′)g(e′) hU,hU U U | C(G0) U U U∈U U U s(e′)=s(e) X X X X p p 8 ADAMRENNIE,DAVIDROBERTSON,ANDAIDANSIMS Since s restricts to a homeomorphism on each U U, we can only have f (e) and f (e′) U U ∈ simultaneously nonzero in the sum on the right-hand side of (2.1) if e = e′. Since f is U real-valued, we have f (e) = f (e), and so U U (θp g)(e) p= h (e)2g(e) = f (e) g(e) = g(e). hU,hU U U UX∈U UX∈U (cid:16)XU (cid:17) This proves that the h constitute a frame. For the final assertion, we calculate U Φ(Id )(v) = (h h )(v) = h (e)h (e) E C(G0) U U U U | U U r(e)=v X X X = f (e) = 1 = vG1 . (cid:3) U | | r(e)=v U r(e)=v X X X 2.1.5. Cuntz–Krieger algebras. As a specific case of the example above, suppose that G = (G0,G1,r,s) is a finite directed graph where G0 and G1 both have the discrete topology. We suppose for simplicity that G has no sources and no sinks. If B is the edge-adjacency matrix of G, then the Cuntz–Pimsner algebra O of the right Hilbert A-module E is the Cuntz– E A Krieger algebra O [35, Example 2, page 193]. If we think of the left Hilbert A-module E as B A arightHilbert Aop moduleEop witheop aop = (a e)op and(eop fop) = (f e) op, then the Cuntz–Pimsner algebra OAop is the·Cuntz–K·rieger algebra|O Agoipven bAy th|e transpose Eop Bt (cid:0) (cid:1) of the matrix B, which is given by the graph Gop defined by reversing the edges of G. 2.1.6. Twisted topological graphs. The following construction is due to Li [27]. Suppose that G = (G0,G1,r,s) is a topological graph. Let N = N : α Λ be an open cover of G1. α Given α ,...,α Λ, write N = n N . A c{ollection∈of fu}nctions 1 n ∈ α1...αn i=1 αi S = s TC(N ,T) : α,β Λ αβ αβ { ∈ ∈ } is called a 1-cocycle relative to N if s s = s on N . αβ βγ αγ αβγ Let C (G,N,S) := x C(N ) : x = s x on N and x x C (E1) c α α αβ β αβ α α c ∈ ∈ n αY∈Λ o For x C (G,N,S) and g G1, we write x(g) for the tuple x(g) , with the convention ∈ c ∈ α α∈Λ that x(g) = 0 when g N . Choose for each g E1 an element α(g) such that g N ; α 6∈ α ∈ (cid:0) (cid:1) ∈ α(g) since the s are circle valued, for x,y C (G,N,S), the map g x(g) y(g) does not αβ c α(g) α(g) ∈ 7→ depend on our choice of the assignment g α(g). Now C (G,N,S) becomes a pre-right- c 7→ Hilbert C (G0)-module under the operations 0 (x a)(g) = x(g) a(s(g)), α α · (x y) (v) = x(g) y(g) , and | A α(g) α(g) s(g)=v X (a x)(g) = a(r(g))x(g) α α · for x,y C (G,N,S), a A, α Λ and v G0. Theorem 3.3 of [27] ensures that the c ∈ ∈ ∈ ∈ completion E(G,N,S) of C (G,N,S) is a right-Hilbert A-bimodule. c THE EXTENSION CLASS AND KMS STATES 9 If r : G1 G0 is a local homeomorphism, then there is a left inner-product on E(G,N,S) → satisfying (x y)(v) = x(g) y(g) , A | α(g) α(g) r(g)=v X which again does not depend on our choice of assignment g α(g). The right action is 7→ adjointable for this left inner-product, and E(G,N,S) is then a bi-Hilbertian A-bimodule. 3. A Kasparov module representing the extension class We now show how to represent the Kasparov class arising from the defining extension of a Cuntz–Pimsner algebra of a bimodule. The easy case turns out to be the SMEB case, which we treat first, since in this case we can also obtain more information in the form of an unbounded representative of the Kasparov module. TheSMEBcasedoesnotimmediatelyshowhowtoproceedinthegeneralcase: thedilation of the representation-mod-compacts of O on the Fock module to an actual representation E of O is easily achieved in the SMEB case by using a two-sided Fock module. E Utilising the extra information coming from the bi-Hilbertian bimodule structure allows us to handle the general case, by constructing an A module with a representation of O . E 3.1. The SMEB case. The following theoremsummarises thesituationwhen Φ(Id ) = 1 . E A The bounded Kasparov module representing the extension in this case is implicit in Pimsner [35], and numerous similar constructions of Kasparov modules associated to circle actions have appeared in [2, 5, 30, 34] amongst others. Similar results for the unbounded Kasparov module were obtained in [15]. The Fock module associated to C∗-bimodules E over a C∗- algebra A is defined as F := E⊗n E n≥0 M with E⊗0 := A, where the internal product E⊗n is taken regarding E as a right A-module with a left action of A. We let [ext] denote the class of the extension 0 End0(F ) T O 0 → A E → E → E → in KK1(O ,End0(E)), and [F ] KK(End0(F ),A) the class of the Morita equivalence. E A E ∈ A E Theorem 3.1. Let E be a SMEB over A. For Z n < 0, define E⊗n := E⊗|n|. Let FE,Z ∋ denote the Hilbert-bimodule direct sum FE,Z := E⊗n, n∈Z M and define an operator N on the algebraic direct sum ∞n=1 nm=−nE⊗m ⊆ FE,Z by Nξ := nξ for ξ E⊗n. There is a homomorphism ρ : OE EndA(FE,Z) such that ρ(se)ξ = e ξ for ∈ → S L ⊗ all e E and ξ E⊗n. The triple ∈ ∈ S (OE,(FE,Z)A,N) is an unbounded Kasparov O –A module that represents the class [ext] [F ] E ⊗End0A(FE) E ∈ KK1(O ,A). E 10 ADAMRENNIE,DAVIDROBERTSON,ANDAIDANSIMS Proof. If E is a SMEB, then the conjugate module E is also a SMEB, and we have E E = A ∼ ⊗ End0(E) = A, and similarly E E = A. This shows that the coefficient algebra A is the A ∼ ⊗A ∼ fixed point algebra for the gauge action, and that the spectral subspaces for the gauge action are full. Then by [5, Proposition 2.9], (OE,(FE,Z)A,N) is an unbounded Kasparov module. The corresponding bounded Kasparov module is determined by the non-negative spectral projection of N, denoted P+, [20, Section 7]. Since P+FE,Z is canonically isomorphic to FE and compression by P implements a positive splitting for the quotient map q : T O , + E E we deduce that (OE,FE,Z,2P+ 1) represents [ext], and hence (OE,FE,Z,N) does to→o. (cid:3) − 3.2. An operator-valued weight. Our next goal is to construct a Kasparov module repre- senting the extension class in the case when E is not a SMEB. To do so, we seek to dilate the Fock representation of T to a representation of O , but we cannot do this using the module E E FE,Z above when E is not a SMEB; the 2-sided direct sum does not carry a representation of O by translation operators. In [35], Pimsner circumvents this problem by enlarging E E to a module E over the core Oγ, and enlarging the Fock module accordingly. This has the ∞ E disadvantage, however, that the resulting exact sequence 0 End0 (F ) T O = O 0 → OγE E∞ → E∞ → E∞ ∼ E → is very different from the sequence 0 End0(F ) T O 0 in which we were → A E → E → E → originally interested. For example, if A = C and E = C2, then End0(F ) = K, whereas A E ∼ EndOγ(FE∞) is Morita equivalent to the UHF algebra M2∞(C). E In this subsection, we show how to dilate the Fock representation without changing coeffi- cientswhenE isafinitelygeneratedbi-Hilbertianbimodulesatisfyingananalytichypothesis, and present some examples of this situation. This will require some set-up building on the tools developed in Section 2. We construct the desired Kasparov module in subsection 3.3. Fix a bi-Hilbertian A-bimodule E, and let e be a frame for the right module E . Given i A { } a multi-index ρ = (ρ ,...,ρ ) we write e = e e for the corresponding element of 1 k ρ ρ1 ⊗···⊗ ρk the natural frame of E⊗k. We define (3.1) eβk = (e e ) = Φ (Id ), A ρ ρ k E⊗k | |ρ|=k X and just write eβ for eβ1. Provided that E is full and finitely generated, eβk is a positive, A invertible and central element of A, [18], so that β A is the logarithm of Φ (Id ). Since k k E⊗k ∈ E will always be clear from context, this justifies our notation (3.2) β := log(Φ(Id )), β := log(Φ(Id )). E k E⊗k We write ρ = ρρ for the decomposition of a multi-index ρ into its initial and final segments, whose lengths will be clear from context. From the formula (e e ) = (e (e e ) e ) A ρ ρ A ρA ρ ρ ρ | | | we see that for 0 n k ≤ ≤ eβk = (e eβn e ) eβn eβk−n. A ρ ρ | ≤ k k |ρ|=k−n X