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APPEARED IN BULLETIN OF THE

arXiv:math/9210227v1 [math.OA] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 292-297VOICULESCU THEOREM, SOBOLEV LEMMA, ANDEXTENSIONS OF SMOOTH ALGEBRASXiaolu WangDedicated to the memory of Xian-Rong WangAbstract. We present the analytic foundation of a unified B-D-F extension functorExtτ on the category of noncommutative smooth algebras, for any Fr´echet operatorideal Kτ .

Combining the techniques devised by Arveson and Voiculescu, we generalizeVoiculescu’s theorem to smooth algebras and Fr´echet operator ideals. A key notioninvolved is τ-smoothness, which is verified for the algebras of smooth functions, via anoncommutative Sobolev lemma.

The groups Extτ are computed for many examples.1. IntroductionFor a compact manifold M, the extensions of C(M) by the compact operatorsK(H) form an abelian group Ext(M), coinciding with the odd K-homology K1(M)[BDF], which can also be defined in terms of the elliptic operators [A1].

This wasa starting point of [K].For a Schatten ideal Lp, the notion of Lp-smooth elements in Ext(M) was intro-duced and studied in [D, DV], and generalized in [S1, G]. Connes constructed theChern character of extensions of smooth algebras by Lp in the cyclic cohomologyof the smooth algebras [C].

When p = 1 it recovers the trace forms in [HH, CP].Today Ext-theory for C∗-algebras has developed into a multifaced field of funda-mental importance in modern analysis, as the meeting ground of classical operatortheory, in particular Toeplitz operators, Wiener-Hopf operators, and noncommuta-tive differential geometry [C], especially pseudodifferential operators, index theory,K-theory, and cyclic homology.While our knowledge of topological Ext-theory is rather complete, a naturaland fundamental problem [A2, p. 9; D, p. 68; HH, p. 236] remains wide open innoncommutative differential geometry, i.e., the formulation and understanding ofthe extension theories of smooth algebras. It would naturally serve as the domainof the Chern character defined in [C].

Since smooth algebras are Fr´echet algebras,it is desirable to have the extension theory constructed for Fr´echet operator ideals.In this note we present the analytical foundation of such a general theory, whichproduces a functor Extτ from the category of smooth algebras to abelian groups,1991 Mathematics Subject Classification. Primary 46L80; Secondary 47B10, 19K33, 47B10,46L87, 46J15, 46M20.Key words and phrases.

Fr´echet algebra, smooth extension, operator ideal, τ-smooth com-pletely positive map, quasi-central approximate identity.Received by the editors March 24, 1992This research was supported in part by U.S. National Science Foundation grant DMS9012753c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2XIAOLU WANGfor any Fr´echet operator ideal Kτ [P; W1, 10.11]. Such a theory will establish aunified framework into which all previous results in this direction fit.There are two well-known technical results forming the cornerstone of Brown-Douglas-Fillmore theory.

One is the celebrated Voiculescu’s theorem [V1], whichgeneralizes Weyl-von Neumann-Berg theorem to C∗-algebras. In [V2] it is extendedto normed ideals and algebras with countable bases.

The other is Stinespring’stheorem [S]. We generalize both theorems to Fr´echet ideals and smooth algebras,answering a question in [D], and we illustrate the theory in the case of smoothmanifolds.2.

Smooth category and Voiculescu’s theoremBy a smooth algebra we mean a Fr´echet ∗-algebra A∞equipped with a norm psuch that p(a∗a) = p(a)2 for all a ∈A∞. Often it is denoted as a pair (A∞, A),where A is the C∗-completion of A∞with respect to p. A prototype is (C∞(M),C(M)) for a compact smooth manifold.The smooth category is the category of all separable smooth algebras, with mor-phisms given by ∗-homomorphisms of Fr´echet ∗-algebras while contractive withrespect to the C∗-norms.There are two key ingredients in our analysis.

Both can be taken for granted forC∗-algebras. One is the existence of quasi-central approximate identity, identifiedin [Ar, V2].

The other is τ-smoothness, which had not appeared in literature untilnow.We say a completely positive map φ: A∞→L(H) is τ-quasi-central, if there isan increasing sequence {km} of finite rank positive operators strongly convergingto 1 such thatlimm→∞[km, φ(a)] = 0in Kτ for every a ∈A∞.Let (A∞, A) be a smooth algebra with a dense ∗-subalgebra A∞0 , which is count-ably generated as a vector space. A completely positive map φ from (A∞, A) intoL(H) is τ-smooth (or Kτ-smooth) with respect to A∞0 , if the following propertyholds:If φ0 is a completely positive map from A into L(H) such that(1)(φ −φ0)(A∞0 ) ⊂Kτ(H),then(φ −φ0)(A∞) ⊂Kτ(H).For (A∞, A) ⊂L(H), if we add (φ −φ0)(A∞∩Kτ(H)) ⊂Kτ(H) in (1), then weobtain the definition of a τ-smoothmod Kτ (or Kτ-smoothmod Kτ) map.A smooth operator algebra (A∞, A) is called τ-smooth (or τ-smooth mod Kτ) ifthe map idA∞is so.

The following is the generalized Voiculescu’s theorem.Theorem 1. Let (A∞, A) be a separable operator algebra τ-smoothmod Kτ inL(H).Let π be a nondegenerate τ-quasi-central representation of (A∞, A) intoL(H) such that π|A∩K(H) = 0.

Then there are unitaries Un : H →H ⊕H such that(U ∗n(a ⊕π(a))Un −a) ∈Kτ(H)for every a ∈A∞,lim(U ∗n(a ⊕π(a))Un −a) = 0in Kτ(H) for every a ∈A∞0 .Restricting the ideal Kτ and A∞to various categories, one recovers all the previ-ous results in this direction. The idea of the proof is an extension of the techniquesinvented in [Ar] and [V2] for Fr´echet algebras.

VOICULESCU THEOREM AND SOBOLEV LEMMA33. Smooth extensionsA τ-smooth extension of (A∞, A) is a pair (π, P), where π is a representation ofA in a Hilbert space H and P is a projection in H such that (i) [π(a), P] ∈Kτ(H)for a ∈A∞, and (ii) Pπ(A)P ∩K(H) = {0}.Define a completely positive map φ(a) := Pπ(a)P, a ∈A.

Then E∞:= φ(A∞)+Kτ is a Fr´echet ∗-algebra with the locally convex final topology induced by themaps i: Kτ →E∞and φ, and (E∞, E) is a smooth operator algebra with a dense∗-subalgebra E∞0 := φ(A∞0 ) + Kf countably generated as a vector space.Two τ-smooth extensions are unitarily equivalent if the two associated com-pletely positive maps on A∞are unitarily conjugate up to Kτ-compact perturba-tion. A τ-smooth extension is degenerated if it is unitarily equivalent to a repre-sentation.Let Extτ(A∞) be the unitary equivalence classes of the τ-smooth extensions ofA∞.

A spatial isomorphism M2(L(H)) ≃L(H) turns Extτ(A∞) into an abeliansemigroup. The quotient abelian monoid modulo the degenerate extensions will bedenoted by Extτ(A∞).We shall denote by Exts,τ(A∞) those represented by τ-smooth completely posi-tive maps.

Replacing Extτ by Exts,τ in the above, we get the submonoid Exts,τ(A∞)of those consisting of τ-smooth completely positive maps. By a refinement of Stine-spring’s theorem [S], we can show that Extτ is a contravariant functor from thecategory of smooth algebras to the category of abelian groups.If τ is finer than τ′, there is a natural transformation ατ,τ ′ from Extτ to Extτ ′.In particular, there is a natural transformation ατ from Extτ to the B-D-F functorExt for any τ. Theorem 1 impliesCorollary.

If every extension in Extτ(A∞) can be represented by a τ-smooth com-pletely positive map and is τ-quasi-central if it is degenerate, then there is a naturalisomorphism of abelian groupsExtτ(A∞) ≃Extτ(A∞).⊔⊓We check the two conditions above for the case of commutative smooth algebras(A∞, A) where A = C(M), for a compact smooth manifold M of dimension n. Wemay assume M is embedded in RN (N ≤2n, by Whitney’s theorem). From thedeep results in [V2], we haveTheorem 2 (Voiculescu).

Notation is as above. All degenerated smooth extensionsof (A∞, A) by LN are LN-quasi-central.For the other condition we haveTheorem 3.

Let M be a compact C(k)-manifold of dimension n. For any Fr´echetoperator ideal Kτ, any completely positive map defining extensions of C(n+2+ε)(M),C(M)) by Kτ is always τ-smooth, for any ε > 0. Here we assume k ≥n + 2 + ε.The key step in the proof is the following noncommutative Sobolev lemma.

Itimplies that if a sequence (am1,...,mn) belongs to a Sobolev space Hs for sufficientlylarge s, then the quantized generalized function determined by (am1,...,mn) con-verges in the τ-smooth quantized Fr´echet algebra.

4XIAOLU WANGLemma. Let (A∞, A) be a commutative smooth algebra with n generators {x1, .

. .

, xn},such that(1) The C∗-norm ∥xj∥≤1, for all j = 1, . .

. , n;(2) Any element in A∞has the formf =X(m1,...,mn)∈Znam1,...,mnxm11.

. .

xmnn ,such thatX(m1,...,mn)∈Znam1,...,mn(|m1| + · · · + |mn|)2 < ∞.For any Fr´echet operator ideal Kτ, let φ be the completely positive map associatedto any τ-smooth extension of (A∞, A). Then φ is τ-smooth.4.

Examples1. Let A = C(X) where X is a compact, second countable, totally disconnectedspace.

Then A has a single generator. If A∞also does, then Exts,τ(A∞) = 0.2.

Let A = A∞= C(S1). There is a representation π: A −→L(H) defining adegenerate τ-smooth extension, which is K-smooth but not L1-smooth as a com-pletely positive map.

Thus ExtL1 is not a compatible functor for the category ofC∗-algebras.3. For any p > 1, there is a degenerate faithful representation of (L1(T ),C∗(T ))defining a Lp-smooth extension, which is not Lp-smooth as a completely positivemap.This shows that if A = C∗(G) for a compact Lie group G, L1(G) is too large asmooth subalgebra.

One needs to take, for example C∞(G) instead.4. For any operator ideal Kτ, we haveExtτ C∞(S1) ≃Ext C(S1) ≃Z.Fix Kτ ⊂Lp for some p ≥1.

It follows from [C] that there is a natural grouphomomorphismchm : Extτ(A∞) −→HC2m+1(A∞),m ≥p,such that chm+1 = S ◦chm. Here HC∗is the cyclic homology.5.

Let D be the unit disc. Then Ext(C(D)) = 0.

However, ExtL1C∞(D) ⊗Ccontains as a direct summand the group HC1(C∞(D)), which is the space of allthe closed de Rham currents on D of dimension 1.Remarks. Since there is no hausdorfftopology on L/Kτ, we abandon the conven-tional formulation of Ext; so a lifting problem does not arise.An attractive perspective of the smooth extension theory is that Extτ(A∞) isa differential invariant for appropriate Kτ and A∞.

In [Km] it is shown that evenin the B-D-F group the class of a smooth extension may depend on the smoothstructure.Details will appear elsewhere (see [W2, W3, W5]). We plan to investigate thetopological aspect of the theory, along with the even degree functors in a futurework.

VOICULESCU THEOREM AND SOBOLEV LEMMA5AcknowledgmentWe would like to thank J. Anderson, L. Brown, A. Connes, R. Douglas, J. Hor-vath, J. Kaminker, and especially J. Rosenberg, and D. Voiculescu, for stimulatingconversations concerning various aspects of this work.References[A1]M. F. Atiyah, Global theory of elliptic operators, Proc. Int.

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6XIAOLU WANG[W5]X. Wang, Smooth K-homology, Chern character, and a noncommutative Sobolev Lemma,in preparation.[W6]X. Wang, Geometric BRST quantization and smooth Toeplitz extensions, in preparation.Department of Mathematics, University of Maryland, College Park, Maryland20742E-mail address: xnw@karen.umd.edu

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