Spectral Invariance of Smooth Crossed

Products, and Rapid Decay Locally Compact

Groups. (Early version of the paper - Only

valid, in general, for a trivial action of G on

A.)

Ronghui Ji and Larry B. Schweitzer

1994

Abstract

We show that all rapid-decay locally compact groups are unimod-

ular and that the set of rapid-decay functions on a locally compact

rapid-decay group with values in a C?-algebra forms a dense and spec-

tral invariant subalgebra of the twisted crossed product C?-algebra.

1

Some generalizations are also obtained. The spectral invariance prop-

erty implies that the K-theories of both algebras are naturally isomor-

phic under inclusion. 1

Contents

0 Introduction 2

1 Definitions and Smooth Crossed Products 8

2 Property RD, Examples, and the Main Theorem 23

3 Proof of Theorem 2.5 37

4 References 50

0 Introduction

The applications of C?-algebras to geometry and index theory, and the com-

putation of the K-theory of C?-algebras, have been the topics of much recent

literature, e. g. [BC, 1988][BCH, to appear] [Bl, 1986] [Co, 1985] [CM, 1990]

1This project is supported in part by the National Science Foundation Grant #DMS

92-04005

2

and the references therein. As an approach to these related problems, one

may adopt the view that when a C?-algebra is accompanied by a dense sub-

algebra of smooth functions, it is reasonable to think of the C?-algebra as

corresponding to a “noncommutative differentiable manifold” , and the dense

subalgebra as “C∞ functions on the manifold” (i.e. C∞(M) if the C?-algebra

is commutative and unital). For instance, the topological information of the

K-theoretic index of a Γ-invariant geometric operator on a covering manifold

M with the covering group Γ can be recaptured by the Chern character of

the index, which lies in the noncommutative deRham theory, i.e., the cyclic

cohomology, of a dense subalgebra A of the stabilization C?r (Γ) ⊗ K of the

reduced group C?-algebra C?r (Γ).

In order that the fundamental pairing of Connes [Co, 1985] between the

K-theoretic index of the geometric operator and the cyclic cohomology of the

dense subalgebra A be a topological invariant of the manifold M = M/Γ,

one must establish that the topological K-theory of A is the same as the K-

theory of C?r (Γ) (see, e.g. [CM, 1990]). One way to do this is to show that A

is spectral invariant in C?r (Γ)⊗K [Bo, 1990], Lemma A.2.1 or [CaPh, 1991],

Appendix and [Sch, 1992]. (Spectral invariance means that the spectrum of

every element of A remains the same in either algebra.)

3

The main task of this paper is to prove spectral invariance for a class of

twisted smooth crossed products S`2(G,A;σ), which occur as smooth dense

subalgebras of the reduced C?-crossed product C?r (G,A;σ). (Here G is a

locally compact group acting on a C?-algebra A, ` is a gauge (or length func-

tion - see Definition 1.1) on the group, and S`2(G,A;σ) is the Frechet algebra

of `-rapidly vanishing L2-functions from G to A, with σ-twisted convolution

multiplication, where σ is a Borel measurable two-cocycle on G (see §1).) The

results of this paper may have potential applications to the Baum-Connes

conjecture with coefficients [BCD, to appear].

The problem of spectral invariance for smooth crossed products, and its

applications, has been looked at before [BC, 1988], Appendix 1 [Bo, 1990]

[Ji, 1992] [Ji, 1993] [Jo, 1989] [Sch, 1993] [Sch, 1993a] [Vi, 1990] [Vi, 1992].

The spectral invariance of the L1 smooth crossed product S`1(G,A) (which by

definition consists of `-rapidly vanishing L1 functions from G to A) is done

for polynomial growth groups in [Sch, 1993a]. There has been a large amount

of literature to determine for which groups G the L1 group algebra L1(G) is

spectral invariant in C?(G) [Pa, 1978], p. 696, p. 728. Such groups are called

hermitian (or symmetric) groups, and include all polynomial growth groups,

the ax+ b group, and many other groups. However, noncompact semisimple

4

Lie groups are never hermitian [Je, 1973]. Hence, in some important cases,

the L1 smooth crossed products S`1(G,A) are not spectral invariant.

If instead of using L1 functions, we use L2-rapidly vanishing functions, we

recover many of these groups. In [PHa, 1988] [Jo, 1990] [Jo, 1989], discrete

groups such as the free groups on n generators, SL(2,Z), and many other

non-hermitian groups, were shown to have the property that the set of L2-

Schwartz functions S`2(G) forms a spectral invariant dense Frechet subalgebra

of the reduced group C?-algebra C?r (G). (A group G together with a gauge

` for which the inclusion S`2(G) ↪→ C?r (G) holds is called rapid decay (RD).

See the beginning of §2 for Examples.) In contrast to hermitian groups,

solvable rapid decay groups must have polynomial growth (see [Jo, 1990],

Corollary 3.1.8 or §2, Example (5) below). In [Jo, 1990] and in [JoVa, 1991],

it was shown that semisimple Lie groups such as SL(2,R) are rapid decay

groups, as are the following classes of groups: a) the groups of isometries

of a Riemannian symmetric space of rank one with noncompact type; b)

the discrete groups acting properly discontinuously with compact quotient

on hyperbolic and geodesic metric spaces in the sense of Gromov; c) any

locally compact unimodular group acting properly on a locally finite tree,

with a finite quotient graph. However, the spectral invariance of S`2(G) in

5

the reduced C?-algebra C?r (G) was not obtained in general.

In this paper, we show that any rapid-decay locally compact group must

be unimodular (Theorem 2.2 below). We then show that for any rapid decay

locally compact group, the group Schwartz algebra S`2(G) is spectral invariant

in the reduced group C?-algebra generalizing Jolissaint’s result in the discrete

case. Moreover, we generalize these results to the case of the twisted smooth

crossed product S`2(G,A;σ), where A is any C?-algebra as mentioned above,

and σ is any Borel measurable two-cocycle on G (see Theorem 2.5 below).

We generalize this result even further, replacing A with a dense Frechet

subalgebra A which is strongly spectral invariant in A (see Definition 1.7).

For example, if G were acting on a locally compact space M , one might

want to replace the C?-algebra A = C0(M) with A = S(M), some version

of Schwartz functions on M (see Example 1.9 and §2, Examples (7)-(10)). If

(G, `) has property RD and the action of G on A is `-tempered (see Definition

1.3), then we show that S`2(G,A;σ) is a spectral invariant dense Frechet

subalgebra of C?r (G,A;σ) (Corollary 2.7). (This is our most general result

on spectral invariance.) The `-temperedness condition places a restriction

on how far away A can be from the C?-algebra A. (Especially if G is a Lie

group which is not Type R - see §2, Example (8).)

6

This replacement ofA withA is one reason for the introduction of the aux-

illiary Banach algebra |C?r |(G,A;σ) and Frechet algebra |C?

r |`(G,A, σ) in §1

(see Definition 1.4). Since S`2(G,A;σ) is not usually contained in L1(G,A;σ)

([Jo, 1990], Theorem 3.1.7), |C?r |(G,A;σ) works in place of the latter. Imi-

tating a theorem for the L1 case ([Sch, 1993a], Theorem 6.7), we show that if

A is strongly spectral invariant in A, then |C?r |`(G,A, σ) is strongly spectral

invariant in |C?r |(G,A;σ) (see Theorem 1.8). (The proof would not work

with spectral invariance in place of strong spectral invariance, which is one

of the reasons for introducing strong spectral invariance.) Since S`2(G,A;σ)

is an ideal in |C?r |`(G,A, σ) (Proposition 1.10, Corollary 2.4), we obtain the

spectral invariance of the former in S`2(G,A;σ).

This allows us to conclude the spectral invariance of S`2(G,A;σ) in the

C?-crossed product from the spectral invariance of S`2(G,A;σ) in the C?-

crossed product (see “Proof of Corollary 2.7 from Theorem 2.5”in §2 below).

The task of §3 is then to obtain the spectral invariance of S`2(G,A;σ). We

introduce the twisted Roe algebra BαG(A) (Definition 3.2) and show that the

C? completion BαG(A), contains (an isomorphic copy of) the reduced C?-

crossed product C?r (G,A;σ) as a fixed point algebra for an action β of G

(see Proposition 3.4). Then we define an ideal J of BαG(A) (Definition 3.8),

7

and an action γ on J involving the gauge ` (Lemma 3.5). We show that

the set of C∞-vectors J∞, when intersected with the fixed point algebra for

β, is none other than S`2(G,A;σ) when (G, `) has property RD (Proposition

3.10). Whence we obtain the spectral invariance of S`2(G,A;σ) in the reduced

C?-crossed product. In fact, since the spectral invariance was obtained by

taking C∞-vectors, we get the stronger result that S`2(G,A;σ) satisfies the

differential seminorm condition (see Definition 1.7, Theorem 2.5, Lemma

3.11, [BlCu, to appear]).

We would like to thank Guoliang Yu, Jonathan Rosenberg, and Theodore

Palmer for helpful comments and suggestions. We also would like to thank

Alain Valette for his kind encouragement and for sending us a copy of his

joint paper with Paul Jollisaint [JoVa, 1991].

1 Definitions and Smooth Crossed Products

Let (A,α,G) be a C?-dynamical system, where G is a locally compact (Haus-

dorff) topological group with identity element e, and A is a C?-algebra on

which G acts via α.

8

Recall that a Borel measurable two cocycle σ on G is a Borel measurable

map σ : G × G → T such that σ(a, b)σ(ab, c) = σ(a, bc)σ(b, c), σ(e, b) =

σ(a, e) = 1, and σ(a, a−1) = 1, for all a, b, c ∈ G. It is well-known that

this last condition can be made within the cohomology class of σ when

the measurability of σ is not required. From these conditions one obtains

the extra condition: σ(a, b) = σ(b−1, a−1). In fact, σ(a, b) = σ(ab, b−1) =

σ(b−1a−1, a) = σ(b−1, a−1).

Let L∞c (G,A) denote the set of compactly supported L∞ functions from

G to A (with compact supports up to null sets). (See [Tr, 1967], §46.1, p.

468, for the definition of L∞(K,A), where K is a compact subset of G. Then

define L∞c (G,A) to be the union (or inductive limit) of the L∞(K,A), where

K ranges over all compact subsets of G.) Then L∞c (G,A) is a ?-algebra under

the σ-twisted convolution product

ϕ ?σ ψ(t) =

∫G

ϕ(s)αs(ψ(s−1t))σ(s, s−1t)ds,

and involution

ϕ?(s) = αs(ϕ(s−1)?)∆(s)−1.

Assume that A is ?-represented faithfully on a Hilbert space H. Then

L∞c (G,A) is ?-represented as a ?-algebra of bounded operators on the Hilbert

9

space L2(G,H) via the formula

ϕ ?σ ξ(t) =

∫G

αt−1(ϕ(s))ξ(s−1t)σ(t−1, s)ds,

where ϕ ∈ L∞c (G,A), and ξ ∈ L2(G,H). The reduced and σ-twisted crossed

product C?-algebra C?r (G,A, α;σ) is defined to be the completion of L∞c (G,A)

in this representation. We will often suppress the α from the notation.

Note that we take the completion of L∞c (G,A) rather than of the com-

pactly supported continuous functions Cc(G,A), because the latter may not

be closed under the σ-twisted convolution if σ is not continuous. However,

since Cc(G,A) is dense in L∞c (G,A) in the C? topology, the completions will

be the same if σ is continuous.

For further details of the reduced crossed product C?-algebra of A by G,

we refer the reader to [Ped, 1979], and to [ZM, 1968] for the σ-twisted reduced

crossed product C?-algebra of A by G, in the case that G is countable and

discrete.

Remark. Throughout this paper, ∆ will be the modular function on G

which satisfies the conventions

∫f(hg)dh = ∆(g)−1

∫f(h)dh,

10

∫f(h−1)dh =

∫∆(h)−1f(h)dh,

for f ∈ Cc(G).

Definition 1.1. A Borel measurable function ` : G→ [0,∞) which satisfies

`(gh) ≤ `(g) + `(h), `(g−1) = `(g), and `(e) = 0 is called a gauge on G.

(Gauges are also called length functions in [Jo, 1990] [Ji, 1992]. See [Prom,

1985], §2 and Example 4.2(2) for more on the terminology.) Gauges are

automatically bounded on compact sets [Sch, 1993b], Theorem 1.2.11. A

gauge `1 strongly dominates another gauge `2(`2 ≤s `1) if there exists d ∈ N

and c > 0 such that

`2(g) ≤ c(1 + `1(g))d, g ∈ G.

The two gauges `1 and `2 are strongly equivalent if `1 ≤s `2 and `2 ≤s `1.

If G is compactly generated, with open and relatively compact generating

set U satisfying e = 1G ∈ U,U = U−1, and ∪∞n=0Un = G, then the word gauge

`U(g) = min{n|g ∈ Un} is a gauge on G. The strong equivalence class of `U

is independent of the choice of generating set U , and the word gauge strongly

dominates every other gauge on G [Sch, 1993b], Theorem 1.1.21. A special

case of this definition is the word length function on a finitely generated

discrete group, where U is a set of generators together with their inverses.

11

We will be considering general gauges ` throughout this paper, and will say

explicitly if ` is to be the word gauge.

(1.2) For G and ` as in (1.1), let S`2(G,F ) be the set of all L2-Schwartz

functions from G into F with respect to the gauge `, where F is a Frechet

space with a given increasing sequence of seminorms {‖ · ‖n}. Therefore,

S`2(G,F ) is the set of square integrable functions ϕ : G→ F which satisfy

|‖ϕ‖|n,k =

(∫G

‖ϕ(g)‖n2(1 + `(g))2kdg

)1/2

<∞,

for all n, k = 0, 1, 2, . . . . Then S`2(G,F ) becomes a Frechet space with the

topology generated by the seminorms {|‖ · ‖|n,k}. It is clear that F can be

any Banach or C?-algebra. In the case when F = B is a Banach algebra, we

will denote the seminorms simply by |‖ϕ‖|k, for ϕ ∈ S`2(G,B). In the case

when F is C, we will simply write S`2(G) for S`2(G,C).

Definition 1.3. By a Frechet algebra we mean a Frechet space (over the com-

plex numbers) with an algebra structure such that multiplication is jointly

continuous [Wa, 1971], Chapter VII. By a Frechet ?-algebra we mean a

Frechet algebra with a continuous involution defined on it. A Frechet al-

gebra A is said to be m-convex if there exists a generating, increasing set of

12

seminorms {‖ · ‖n} which are submultiplicative. That is, for each n,

‖ab‖n ≤ ‖a‖n · ‖b‖n, a, b ∈ A.

Let A be a Frechet algebra with a continuous action α of G. We say that the

action α is `-tempered if for any m ∈ N, there exists p ≥ m, q ∈ N, and c > 0,

such that

‖αg(a)‖m ≤ c(1 + `(g))q‖a‖p,

for all g ∈ G, and a ∈ A. We say that α is m-`-tempered if there exists a

family of submultiplicative seminorms {‖ · ‖n} topologizing A such that for

any m ∈ N, there exists q ∈ N, c > 0, such that

‖αg(a)‖m ≤ c(1 + `(g))q‖a‖m,

for all g ∈ G, and a ∈ A. (This condition insures the m-convexity of the

crossed product. See [Sch, 1993b], §3 and Theorem 1.6 below.)

If A is a Frechet ?-algebra with `-tempered action of a unimodular group

G by ?-automorphisms, then the involution ϕ?(g) = αg(ϕ(g−1)?), defined at

the beginning of §1, gives a conjugate linear automorphism of the Frechet

space S`2(G,A).

Definition 1.4. Let (G, `) be as in Definition 1.1, and let E be any Frechet

13

space, topologized by seminorms {‖ · ‖n}. Define seminorms on L∞c (G,E) by

‖ϕ‖d,n = ‖ϕd,n‖C?r (G), (1.5)

where the expression on the right is the C?r (G)-norm of the function ϕd,n :

g → (1 + `(g))d‖ϕ(g)‖n. (We define L∞c (G,E) when E is a Frechet space in

exactly the same way we did at the beginning of §1 when E was a C?-algebra.

The treatment in [Tr, 1967], §46.1, p. 468 we referred to is in fact for Frechet

spaces.) Let |C?r |`(G,E) denote the completion of L∞c (G,E) in the topology

determined by these seminorms. (Another approach would be to take the

possibly weaker seminorms ‖ϕd,n‖C?r (G;σ). We choose not to let the topology

of |C?r |`(G,E) depend on σ.) If G is discrete, |C?

r |`(G,E) consists precisely

of the functions ϕ : G → E for which the seminorms (1.5) are finite. If

` = 0 is the trivial gauge, we write |C?r |(G,E). This is a Banach space if

E is Banach. It is easy to check that if T : E → F is a continuous map,

then there is a natural continuous linear map T : |C?r |`(G,E)→ |C?

r |`(G,F )

defined by T (ϕ)(g) = T (ϕ(g)) for ϕ ∈ L∞c (G,E). Also, T has a dense image

if T does.

Theorem 1.6. LetA be an (m-convex) Frechet algebra with (m)-`-tempered

action of G by Frechet algebra automorphisms. Let σ be a two cocycle on G

14

with values in T. Then |C?r |`(G,A) is an (m-convex) Frechet algebra under

the σ-twisted convolution, which we denote by |C?r |`(G,A;σ). If A is a

Frechet ?-algebra, and G acts by ?-automorphisms, then |C?r |`(G,A;σ) is a

Frechet ?-algebra.

Proof: Let ϕ, ψ ∈ L∞c (G,A). Recall the twisted convolution product

ϕ ?σ ψ(h) =

∫ϕ(k)αk(ψ(k−1h))σ(k, k−1h)dk.

For convenience, replace ` with 1 + `, so that ` is submultiplicative and

` ≥ 1. Let ϕd,m denote the function g → `(g)d‖ϕ(g)‖m as in Definition 1.4.

If ξ ∈ L2(G), we have

|(ϕ ?σ ψ)d,m ? ξ(g)| ≤∫`(h)d‖ϕ ?σ ψ(h)‖m|ξ(h−1g)|dh

=

∫`(h)d

∥∥∥∥∫ ϕ(k)αk(ψ(k−1h))σ(k, k−1h)dk

∥∥∥∥m

|ξ(h−1g)|dh

≤ c

∫ ∫ϕd+q,r(k)ψd,p(k

−1h)|ξ(h−1g)|dkdh,

where we used the `-temperedness and the fact that A is a Frechet algebra

in the last step. Now take the L2(G) norm of the whole thing, and use the

definition of the C?r (G) norm to get

‖ϕ ?σ ψ‖d,m = ‖(ϕ ?σ ψ)d,m‖C?r (G) ≤ c‖ϕ‖d+q,r‖ψ‖d,p.

Hence, |C?r |`(G,A) is a Frechet algebra under the σ-twisted convolution.

15

The m-convexity statement follows from a similar estimate, but with a

convolution product of n elements ϕ1, . . . ϕn of L∞c (G,A) instead of just ϕ

and ψ. See [Sch, 1993b], proof of Theorem 3.1.7.

If A is a Frechet ?-algebra, we have

|(ϕ?)d,m ? ξ(g)| ≤∫

(ϕ?)d,m(h)|ξ(h−1g)|dh

≤ c

∫∆(h)−1ϕd+q,p(h

−1)|ξ(h−1g)|dh,

where we used the definition of ϕ?, the `-temperedness, and `(h) = `(h−1).

Taking the L2(G)-norm, we get

‖ϕ?‖d,m ≤ c‖(ϕd+q,p)?‖C?r (G) = c‖ϕd+q,p‖C?r (G) = c‖ϕ‖d+q,p.

Therefore, |C?r |`(G,A;σ) is a Frechet ?-algebra. �

Definition 1.7. Let ι : A → A be an algebra homomorphism of an algebra

A to a Banach algebra A, with dense image. (Assume that ι(1A) = 1A if A

is unital.) We say that A is spectral invariant (SI) in A if every a ∈ A is

invertible in A if and only if ι(a) is invertible in A [Sch, 1992] [Bo, 1990],

Appendix. It follows that A is SI in A if and only if Mn(A) is SI in Mn(A).

If A is a Frechet algebra, we require the map ι : A → A to be continuous.

(In this case, if ι is injective (as it usually will be in this paper), spectral

16

invariance is equivalent to Mn(A) being closed under the holomorphic func-

tional calculus in Mn(A) for each n [Sch, 1992], Corollary 2.3, Theorem 2.1.

In the terminology of [Ji, 1992], this means that A is smooth in A.)

Let ι : A → A be a continuous algebra homomorphism of a Frechet al-

gebra A to a Banach algebra A, with dense image. Then A satisfies the

differential seminorm condition (DSC) in A [BlCu, to appear], [Sch, 1993a]

if there exists a family of seminorms {‖ · ‖m}∞m=0 for A and a constant c > 0

such that

‖ab‖m ≤ c∑i+j=m

‖a‖i‖b‖j, a, b ∈ A,

where ‖ · ‖0 = ‖ι(·)‖A is given by the Banach algebra norm on A. (The

notation∑

i+j=m is short for∑m

i=0, with j = m − i. ) We say that A is

strongly spectral invariant (SSI) in A [Sch, 1993a], §1, [Sch, 1993] if for every

m, there is some pm ≥ m, c > 0, and Dm > 0 such that

‖a1 · · · an‖m ≤ cnDm

∑‖a1‖k1 · · · ‖an‖kn ,

for all n-tuples a1, . . . , an ∈ A and all n, where the sum is over those k′is such

that∑n

i=1 ki ≤ pm. With this definition we have the implications:

DSC ⇒ SSI ⇒ SI.

([Sch, 1993a], Proposition 1.7, Theorem 1.17.)

17

The following theorem is similar to [Sch, 1993a], Theorem 6.7, except

with |C?r |` and |C?

r | in place of L`1 of L1 respectively.

Theorem 1.8. Let ι : A → A be a continuous algebra homomorphism of a

Frechet algebra A into a Banach algebra A, with dense image. Assume that

the action of G on A and A commutes with i, and that G acts isometrically

on A.

a) If the action of G on A is `-tempered, and A is strongly spectral

invariant in A, then |C?r |`(G,A;σ) is strongly spectral invariant in the Banach

algebra |C?r |(G,A;σ).

b) If A satisfies the differential seminorm condition in A (for seminorms

{‖ · ‖n} of A), and G acts isometrically on A with respect to the seminorms

{‖ · ‖n}, then |C?r |`(G,A;σ) satisfies the differential seminorm condition in

|C?r |(G,A;σ).

Remark. 1) One reason we allow the morphism ι : A → A to be non-

injective, is to avoid the question of whether ι : |C?r |`(G,A;σ)→ |C?

r |(G,A;σ)

is injective.

2) A Frechet algebra which is strongly spectral invariant in a Banach alge-

bra must bem-convex [Sch, 1993a], Proposition 1.7. ThusA and |C?r |`(G,A;σ)

are necessarily m-convex Frechet algebras in the above theorem.

18

Proof: For strong spectral invariance, imitate the proof of [Sch, 1993a],

Theorem 6.7, using |C?r | in place of L1.

We prove b). Assume that A satisfies DSC in A and that the action of

G on A is isometric. Let c > 0 be such that

‖ab‖n ≤ c∑i+j=n

‖a‖i‖b‖j, a, b ∈ A,

where ‖ · ‖0 = ‖ι(·)‖A is given by the norm on A. Let ϕ, ψ be elements of

L∞c (G,A). Define seminorms on |C?r |`(G,A;σ) by

‖ϕ‖n =∑i+j=n

1

i!‖ϕ‖i,j,

where ‖ · ‖i,j are the usual seminorms on |C?r |`(G,A;σ) (1.5), but with `(g)d

in place of (1 + `(g))d. Then ‖ · ‖0 = ‖ι(·)‖0,0 is given by the norm on the

Banach algebra |C?r |(G,A;σ). We have

‖ϕ ?σ ψ‖i,j =

∥∥∥∥`(·)i(∥∥∥∥∫ ϕ(h)αh(ψ(h−1·))σ(h, h−1·)dh∥∥∥∥j

)∥∥∥∥C?r (G)

.

Proceeding as in the proof of Theorem 1.6, and using the DSC for A and the

19

isometric action of G, we then have

‖ϕ ?σ ψ‖i,j ≤ c∑p+q=j

∥∥∥∥`(·)i ∫ ‖ϕ(h)‖p‖ψ(h−1·)‖qdh∥∥∥∥C?r (G)

≤ c∑p+q=j

(∑k+l=i

i!

k!l!‖ϕk,p‖C?r (G)‖ψl,q‖C?r (G)

)= c

∑p+q=j

∑k+l=i

(k + l)!

k!l!‖ϕ‖k,p‖ψ‖l,q,

where the second step used `(g) ≤ `(h) + `(h−1g). Therefore

‖ϕ ?σ ψ‖n ≤ c∑

k+l+p+q=n

1

(k + l)!

(k + l)!

k!l!‖ϕ‖k,p‖ψ‖l,q

= c∑

k+l+p+q=n

1

k!l!‖ϕ‖k,p‖ψ‖l,q = c

∑i+j=n

‖ϕ‖i‖ψ‖j.

Thus |C?r |`(G,A;σ) satisfies DSC in |C?

r |(G,A;σ). This proves Theorem 1.8.

�

(1.9) Example 1) Let A = C∞(T), A = C(T) with pointwise multiplication.

Let G = Z act by an irrational rotation. The action is isometric on both

algebras (where the second algebra is topologized by the sup norm, and the

first by the sup norms of derivatives), so |C?r |`(Z, C∞(T)) satisfies the DSC

in the Banach algebra |C?r |(Z, C(T)), if ` is any gauge on Z.

2) Let A = S(R), the standard Schwartz functions on R, and let A =

C0(R) with pointwise multiplication. Then A satisfies DSC in A. Let G = Z

act by translation. The action on A is not isometric, but is tempered with

20

respect to the word gauge `(n) = |n|. So |C?r |`(Z,S(R)) is SSI in Banach

algebra |C?r |(Z, C0(R)).

For any Frechet space E, recall from (1.2) the definition of S`2(G,E), the

`-rapidly vanishing L2-functions on G with values in E.

Proposition 1.10.a) If G is discrete, then |C?r |`(G,E) ↪→ S`2(G,E) is a

continuous inclusion of Frechet spaces.

b) Assume that a locally compact group G acts `-temperedly on a Frechet

algebra A. If E is a left Frechet A-module, then S`2(G,E) is a left Frechet

|C?r |`(G,A;σ)-module, with the action given by the same formula as the left

regular representation:

ϕ ?σ ξ(t) =

∫G

αt−1(ϕ(s))ξ(s−1t)σ(t−1, s)ds,

where ϕ ∈ L∞c (G,A), ξ ∈ S`2(G,E).

Remark 1.11. If G is not discrete, then |C?r |`(G) may not be contained in

S`2(G). For example, for G = T, the circle group, the word gauge ` is given by

`(g) = 1 if g 6= 1T, `(g) = 0 if g = 1T. Then S`2(G) = L2(G) while |C?r |`(G)

contains L1(G) but L2(G) does not. Therefore, |C?r |`(G) is not contained in

S`2(G).

21

Proof of Proposition (1.10): If G is discrete, let δe be the step function

at e = 1G. Then δe ∈ L2(G) and for ϕ ∈ S`2(G,E)

‖ϕ‖d,m ≥ ‖ϕd,m ? δe‖2 = ‖ϕd,m‖2 = |‖ϕ‖|d,m,

for all d and all m, where |‖ϕ‖|d,m is the seminorm on S`2(G,E) as defined in

(1.2). This proves a).

If ϕ ∈ L∞c (G,A), ξ ∈ S`2(G,E), then there exist c > 0, and p, q, r ∈ N

such that

| ‖ϕ ?σ ξ‖|d,m

=

{∫G

(∥∥∥∥∫G

αg−1(ϕ(h))ξ(h−1g)σ(g−1, h)dh

∥∥∥∥m

)2

(1 + `(g))2ddg

}1/2

≤ c

{∫G

(∫G

‖ϕ(h)‖p‖ξ(h−1g)‖r(1 + `(g))d+qdh

)2

dg

}1/2

≤ c

{∫G

(∫G

‖(1 + `)d+qϕ(h)‖p‖(1 + `)d+qξ(h−1g)‖rdh)2

dg

}1/2

≤ c‖ϕd+q,p‖C?r (G)‖ξd+q,r‖2 = c‖ϕ‖d+q,p|‖ξ‖|d+q,r.

�

22

2 Property RD, Examples, and the Main The-

orem

Definition 2.1. Let G be a locally compact group with gauge `. We say

that (G, `) has property RD (or is rapid decay) if the L2-Schwartz functions

S`2(G) are contained in C?r (G) [Jo, 1990], 1.2. If we simply say that G has

property RD, this means that there exists a gauge ` on G such that (G, `)

has property RD. Note that if (G, `1) has property RD, and `1 is strongly

dominated by another gauge `2, then (G, `2) has property RD. Hence if G

has property RD and is compactly generated, then (G, `U) has property RD,

where `U is the word gauge on G (see Definition 1.1).

Example (1). Let G be a discrete group. According to [Jo, 1990], G has

property RD if G has polynomial growth [Gr, 1981] or if G is a hyperbolic

group. Moreover, the word hyperbolic groups of Gromov [Gr, 1987] are

RD-groups [PHa, 1988]. In [Jo, 1990], Jolissaint also showed that the amal-

gamated free product of two RD-groups by a finite subgroup is once again an

RD-group; and in [JoVa, 1991], Jolissaint and Valette showed that a discrete

group acting properly discontinuously with compact quotient on a hyperbolic

and geodesic metric space in the sense of Gromov is rapidly decaying (this

23

class is bigger than the word hyperbolic groups). This provides a good class

of discrete RD-groups. See also [Jo, 1990], Example 2.1.11 (braid group), and

for non-RD discrete groups see [Jo, 1990], Corollary 3.1.9 (SL(n,Z), n ≥ 3),

Example 1.2.8.

Example (2). Let G = F∞ be the free group on countably infinitely many

generators. Then G is not compactly generated, so there is no word gauge on

G. (Moreover, there is no gauge that dominates every other gauge on G [Sch,

1993b], Example 1.1.15, Theorem 1.1.21.) It is well known that there is an

injective group homomorphism of G into F2, the free group on two generators.

Let ` be the word gauge on F2. Then, as we noted in Example (1), (F2, `)

has property RD. Since G is a subgroup, it follows that (G, `|G) also has

property RD (by an easy argument, or [Jo, 1990], Proposition 2.1.1). Note

that this is one reason we must include gauges other than the word gauge in

the definition of property RD.

Example (3). Since SL(2,R) has real rank one, it has property RD by [Jo,

1990], Corollary 1.4 of the Appendix. Examples of this kind can also be found

in [JoVa, 1991], in which the authors proved more generally that the group

of isometries of a Riemannian symmetric space of rank 1 of noncompact type

24

is rapidly decaying.

Example (4). In [Vi, 1992] and [Vi, 1990], Proposition 28, Theorem 29,

there are many examples of Lie groups which possess properties similar to

property RD.

Example (5). Any locally compact group of polynomial growth is an RD-

group with respect to the word gauge. (Polynomial growth means that G is

compactly generated with generating set U , and that the Haar measure |Un|

is bounded by a polynomial in n.) Moreover, an amenable locally compact

group has property RD if and only if it has polynomial growth. (Proof:

The same arguments as Theorem 3.1.7 and Corollary 3.1.8 of [Jo, 1990] can

be carried over in the locally compact case without change.) Note that all

solvable locally compact groups are amenable [Pa, 1978], p. 729.

The following is an explicit example of a non-RD, connected Lie group.

Example (6). Let G be the ax+ b group

{ea b

0 1

∣∣∣∣ a, b ∈ R

}.

We show that G does not have property RD. (In contrast, L1(G) is spectral

invariant in C?r (G)[Pa, 1978], p. 696 or Example 21.) Note that the Haar

25

measure on G is given by e−adadb, where dadb is Lebesgue measure on R.

The word gauge on G is strongly equivalent to the gauge

τ(g) = ln (1 + e|a| + e−a|b|+ |b|),

[Sch, 1993b], Example 1.6.1.

Let ρ : R2 → R+ be defined by

ρ(a, b) =ea/2e−|a|/10

(1 + |b|)3/4.

Then ρ ∈ Sτ2 (G), since

|‖ρ‖|2k =

∫R

∫Rτ(g)2k

(ea/2e−|a|/10

(1 + |b|)3/4

)2

e−adadb

≤∫ ∫

e−|a|/5

(1 + |b|)3/2polyk(|a|, ln (1 + |b|)) dadb ≤ Ck <∞.

We show that ρ ? ρ 6∈ L2(G) by showing that ρ ? ρ(a, b) does not converge for

any (a, b) ∈ R2 for which a < 0. (Here ? is meant to denote the left regular

representation on L2(G).) In fact,

26

ρ ? ρ(a, b) =

∫ ∫ρ(u, v)ρ(a− u, e−u(b− v))e−ududv

=

∫ ∫eu/2e−|u|/10

(1 + |v|)3/4

e(a−u)/2e−|a−u|/10

(1 + |e−u(b− v)|)3/4e−ududv

≥ ea/2∫ b

−∞

∫ a

−∞

eu/10

(1 + |v|)3/4

e(u−a)/10

e−3u/4(1 + |b− v|)3/4e−ududv

= e2a/5

∫ b

−∞

1

(1 + |v|)3/4(1 + |b− v|)3/4dv ·

∫ a

−∞e−u/20du.

Since∫ a−∞ e

−u/20du does not converge for any a < 0, ρ ? ρ is not defined, and

so ρ ? ρ 6∈ L2(G). Hence ρ /∈ C?r (G), and G does not have property RD.

This example is a special case of the following theorem (and also of the

previous example).

Theorem 2.2. Assume that G is a locally compact group, which is not

unimodular, and let ` be any gauge on G. Then (G, `) does not have property

RD. Moreover, (G, `) cannot be made to have property RD by changing the

convention on the involution to be ϕ?(g) = ϕ(g−1)∆(g)−a for some a ∈ R

other than a = 1.

Proof: We have two proofs of this theorem. The simpler one is similar to

the argument in [HR, 1979], (20.34) p. 308. For the reader’s convenience, we

will give a complete account of this.

Since G is non-unimodular, there exists an a ∈ G such that ∆(a−1) ≥ 2.

27

Let U, V be symmetric neighborhoods of the identity element e ∈ G, so that

U is relatively compact, U2 ⊂ {x ∈ G : 1/2 < ∆(x) < 2} and V 2 ⊂ U . Note

that since gauges are automatically bounded on compact subsets of G [Sc4],

Theorem 1.2.11, we may assume c > 0 is such a bound for ` on U .

Now the sequence of subsets {akU}∞k=−∞ are pairwise disjoint. For k =

1, 2, ..., choose symmetric neighborhoods Wk of e, such that Wk ⊂ V and the

Haar measure λ(Wk) = 2−k. (This can be done since G is non-atomic).

Let nk = 3k + 2. Set W = ∪∞k=1ankWk and A = ∪∞k=1a

nkUak. Then

λ(W ) =∑∞

k=1 λ(Wk) =∑∞

k=1 2−k = 1.

Define ϕ = χW , and f = ∆−1/2χA−1 . We show that both ϕ and f are in

S`2(G), but f ? ϕ /∈ L2(G).

Thus, G cannot be rapidly decaying. In fact, for any positive integer m,

∫|ϕ(g)|2(1 + `(g))2mdg =

∫χW (g)(1 + `(g))2mdg

=∞∑k=1

∫ankWk

(1 + `(g))2mdg

≤∞∑k=1

∫ankWk

(1 + nk`(a) + c)2mdg

=∞∑k=1

2−k(1 + nk`(a) + c)2m <∞,

28

since nk = 3k + 2. Similarly,

∫|f(g)|2(1 + `(g))2mdg =

∫∆(g)−1χA−1(g)(1 + `(g))2mdg

=

∫∆(g)−1χA(g−1)(1 + `(g))2mdg

=

∫A

(1 + `(g))2mdg

≤∞∑k=1

λ(ankUak)(1 + c+ (nk + k)`(a))2m

=∞∑k=1

λ(U)∆(ak)(1 + c+ (4k + 2)`(a))2m

≤∞∑k=1

λ(U)2−k(1 + c+ (4k + 2)`(a))2m <∞.

Now we prove that f ? ϕ /∈ L2(G). Let

I(x) = f ? ϕ(x) =

∫f(g)ϕ(g−1x)dg

=

∫∆(g)−1/2χA−1(g)χW (g−1x)dg

=

∫∆(g)−1/2χA(g−1)χW (g−1x)dg

=

∫∆(g)−1/2χA(g)χW (gx)dg

=

∫∆(ux−1)−1/2χA(ux−1)χW (u)∆(x)−1du

=

∫∆(x)−1/2∆(u)−1/2χA(ux−1)χW (u)du.

For u ∈ ankWk and x ∈ a−kV , we have ux−1 ∈ A. Therefore,

I(x) ≥∫ankWk

∆(x)−1/2∆(u)−1/2du ≥ 2kλ(Wk) = 1.

29

The last inequality can be estimated as follows. Let x = a−kv, u = ankwk,

where v ∈ V,wk ∈ Wk. Then

∆(x)−1/2∆(u)−1/2 = ∆(a−kv)−1/2∆(ankwk)−1/2 ≥ 1

2∆(a)(−nk+k)/2 ≥ 2k.

Since λ(∪∞k=1a−kV ) =∞, f ? ϕ /∈ L2(G).

This completes the proof of the first statement. The second can be proved

by modifying the proof above. �

Proposition 2.3. Assume that (G, `) has property RD. If E is any Frechet

space, then we have a continuous inclusion S`2(G,E) ↪→ |C?r |`(G,E) of Frechet

spaces with dense image. If G is discrete, the map is an isomorphism.

Proof: We have

‖ϕ‖d,m = ‖ϕd,m‖C?r (G) ≤ c|‖ϕd,m‖|p = c|‖ϕ‖|d+p,m

for some p ∈ N and c > 0. So the inclusion map is well defined and

continuous. If G is discrete, we saw in Proposition 1.10 a) above that

|C?r |`(G,E) ↪→ S`2(G,E) with continuous inclusion. �

Corollary 2.4. IfA is an [m-convex] Frechet (?)-algebra with [m]-`-tempered

action ofG by (?)-automorphisms, and (G, `) has property RD, then S`2(G,A)

is a dense left [m-convex] Frechet (?-)ideal in the [m-convex] Frechet (?-

)algebra |C?r |`(G,A;σ). (The two are equal if G is discrete.) In particular,

30

S`2(G,A) is an [m-convex] Frechet (?)-algebra under the σ-twisted convolu-

tion.

Proof: If G is discrete, S`2(G,A) ∼= |C?r |`(G,A;σ) as Frechet spaces (Propo-

sition 2.3), and the theorem follows from the corresponding theorem for

|C?r |`(G,A;σ) (Theorem 1.6). In general, S`2(G,A) is a dense subset of

|C?r |`(G,A;σ) with continuous inclusion by Proposition 2.3, and is a left

Frechet |C?r |`(G,A;σ)-module by Proposition 1.10 b). (Recall that as a

Frechet space, |C?r |`(G,A;σ) does not depend on σ (see Definition 1.4 above).

Thus Proposition 2.3 still gives a continuous inclusion.) We want S`2(G,A) to

be a left ideal in |C?r |`(G,A;σ), or in other words to be a left |C?

r |`(G,A;σ)-

module when the action is given by the multiplication formula:

ϕ ?σ ψ(t) =

∫G

ϕ(s)αs(ψ(s−1t))σ(s, s−1t)ds.

This leads to only a slight modification of the proof of Proposition 1.10 b).

We obtain

|‖ϕ ?σ ψ‖|d,m ≤ c‖ϕ‖d+q,n|‖ψ‖|d+q,r, ϕ, ψ ∈ S`2(G,A).

Since ‖ϕ‖d+q,n ≤ c|‖ϕ‖|d+q+p,n by Proposition 2.3, S`2(G,A) is a Frechet

algebra under convolution.

31

Them-convexity statement follows easily from them-convexity of |C?r |`(G,A;σ)

(Theorem 1.6), using the condition for m-convexity given in [Sch, 1993b],

Theorem 3.1.4. Since G has property RD, G is unimodular (Theorem 2.2),

so the involution ϕ?(g) = αg(ϕ(g−1)?) is well-defined and continuous on

S`2(G,A) as we noted in Definition 1.3, when A is a Frechet ?-algebra and G

acts by ?-automorphisms. �

Notation: To emphasize the role of σ in the product of elements in S`2(G,A),

we will denote this algebra by S`2(G,A;σ). If σ is cohomologous (via a Borel

one-cochain) to the trivial two-cocycle σ0, then S`2(G,A;σ) is isomorphic to

S`2(G,A;σ0). In this case, S`2(G,A) will be used to simplify the notation.

The following theorem and corollary are the main results of the paper.

Theorem 2.5. If (G, `) has property RD, and G acts on a C?-algebra A,

then S`2(G,A;σ) satisfies the differential seminorm condition in C?r (G,A;σ).

In particular, S`2(G,A;σ) is a spectral invariant dense Frechet subalgebra of

C?r (G,A;σ).

Remark 2.6. a) This theorem is also true in general when the two-cocycle

σ has values in the unitary group of the center of the multiplier algebra of

A. All the set-up and proofs go through without any change. However, we

32

will not pursue this generality for the sake of simplicity.

b) If the theorem holds, then C?r (G,A;σ) is necessarily the enveloping

C?-algebra of S`2(G,A;σ) [Bl, 1986], 3.1.3, 3.1.5.

c) The Frechet algebra S`2(G,A;σ) in Theorem 2.5 is necessarily m-

convex, since the DSC implies m-convexity [Sch, 1993a], Proposition 1.7.

It is also a Frechet ?-algebra by Corollary 2.4. (One could also obtain the

m-convexity using Corollary 2.4, since the action of G on any C?-algebra is

isometric and so m-`-tempered.)

Corollary 2.7. Let A be a dense Frechet subalgebra of a C?-algebra A.

Assume that G acts continuously on A and A, and that the action is `-

tempered onA. If (G, `) has property RD andA is strongly spectral invariant

in A, then S`2(G,A;σ) is a spectral invariant dense Frechet subalgebra of the

C?-algebra C?r (G,A;σ).

Remark 2.8. Under the hypotheses of Corollary 2.7, the Frechet algebras

A and S`2(G,A;σ) are necessarily m-convex, since SSI implies m-convexity

[Sch, 1993a], Proposition 1.7.

Proof of Corollary 2.7 from Theorem 2.5: If A is SSI in A, then

both S`2(G,A;σ), and S`2(G,A;σ) are SSI in the Banach algebra |C?r |(G,A;σ)

33

by Theorem 1.8. Thus both are SI in |C?r |(G,A;σ) [Sch, 1993a], Theorem

1.17, and S`2(G,A;σ) is spectral invariant in S`2(G,A;σ). By Theorem 2.5,

S`2(G,A;σ) is spectral invariant in C?r (G,A;σ), and so is S`2(G,A;σ). �

We give some examples to illustrate the main theorems.

Example (7). Let G be any word hyperbolic discrete group with word

gauge ` [Gr, 1987]. Let M be a locally compact space on which G acts.

(For example, we could let M = ∂G, the boundary, and by [Gr, 1987] M is

compact.) Then G acts on C0(M) (continuous functions on M vanishing at

infinity) by translation, and S`2(G,C0(M)) satisfies the DSC in C?r (G,C0(M))

by Theorem 2.5. Alternatively, let G = SL(2,R) act on the boundary of the

hyperbolic plane M = ∂H2. Then we get the same result.

We give an example which illustrates how the `-temperedness condition

can be restrictive.

Example (8). Let G = SL(2,R) act on the hyperbolic plane H2, and let `

be the word gauge on G. By Theorem 2.5, we know that S`2(G,C0(H2)) is

spectral invariant in C?r (G,C0(H2)), by the DSC. We would like to replace

C0(H2) with some smooth subalgebra S(H2) of functions on H2, and apply

Corollary 2.7 to see that S`2(G,S(H2)) is spectral invariant in C?r (G,C0(H2)).

34

Note, however, that Corollary 2.7 requires that the action of G on S(H2) be

`-tempered. If we take S(H2) to be rapidly vanishing continuous functions on

H2 (where rapidly vanishing means with respect to the scale on H2 inherited

from the gauge ` on G), then the `-temperedness condition would be met,

and we can apply Corollary 2.7. On the other hand, if one tries to let S(H2)

be G-differentiable functions, one runs into problems. We usually obtain the

`-temperedness on sets of C∞-vectors from [Sch, 1993b], Theorem 4.6, which

assumes that ` bounds Ad. But for ` to bound Ad, it is necessary that G

be a Type R Lie group [Sch, 1993b], Theorem 1.4.3. This is not the case

for G = SL(2,R). So one must be satisfied with using for S(H2) algebras of

continuous or rapidly vanishing functions, at least without further analysis.

Next, let M = H2×R. Let G act on the first factor as usual, and act triv-

ially on the copy of R. Then we may take S(M) to be C0(H2)⊗C∞0 (R) (dif-

ferentiable in the direction transverse to the action), and the `-temperedness

condition is satisfied.

The strong spectral invariance of S(M) in C0(M) is proved in [Sch, 1993a],

Theorem 2.2, for all the examples mentioned. Also, note that one reason for

the `-temperedness condition in Corollary 2.7 is to insure that S`2(G,A;σ) is

a Frechet algebra under convolution in the first place.

35

For Type R Lie groups, the `-temperedness condition is not so restrictive:

Example (9). Let H be a compactly generated polynomial growth Type R

Lie group (see [Sch, 1993b], §1.4 - 1.5), and let G and K be closed subgroups

of H. (For example, H could be any connected, simply connected nilpotent

Lie group.) Let ` be the word gauge on H. Define a scale ρ on H/K by

ρ([h]) = infk∈K

`(hk), (∗)

and let A = S(H/K) denote the ρ-rapidly vanishing H-differentiable func-

tions on H/K with pointwise multiplication [Sch, 1993b], §5. Let G act by

left translation on A. Then the action of G on A is `-tempered [Sch, 1993b],

Corollary 1.5.12 and Theorem 4.6 (using Type R implies that ` bounds Ad),

so by Corollary 2.7 we know that S`2(G,S(H/K)) is a spectral invariant

dense Frechet subalgebra of C?r (G,C0(H/K)). We could have instead let

A = S(H/K) consist of G-differentiable functions instead of H-differentiable

functions, in which case we would obtain a similar result. (Then A is not

a nuclear Frechet space.) These examples can just as well be done with L1

smooth crossed products [Sch, 1993a], Examples 6.26-7, 7.20, because G has

polynomial growth [Sch, 1993b], Corollary 1.5.11.

36

We combine the two previous examples.

Example (10). Let G1 = SL(2,R) with word gauge `1. Let G2 be a

polynomial growth Type R Lie group with word gauge `2, and let K be a

closed subgroup of G2. Let G = G1×G2 act on H2×G2/K via the action of

G1 on the hyperbolic plane H2, and of G2 on the homogeneous space G2/K.

Let ρ1 : H2 → [0,∞) be defined using `1 as in (∗), and let ρ2 : G2/K → [0,∞)

be defined using `2 as in (∗). Let S(M) = S(H2 ×G2/K) consist of ρ1 + ρ2-

rapidly vanishing continuous functions on M , which are also differentiable in

the G2-direction. Then G acts `1 + `2-temperedly on S(M), and Corollary

2.7 applies with gauge `((g1, g2)) = `1(g1) + `2(g2) on G.

3 Proof of Theorem 2.5

We prove Theorem 2.5 in a series of definitions and lemmas.

Definition 3.1. If A is any C?-algebra, then L2(G,A) is a Hilbert module

over the ?-algebra L∞c (G,A) with the action defined by

ϕ ?σ ξ(g) =

∫αg−1(ϕ(h))ξ(h−1g)σ(g−1, h)dh.

The operator norm on L∞c (G,A) is then a C?-norm, and we let C?h(G,A;σ)

denote the completion. Note that if A = C, then C?h(G;σ) = C?

r (G;σ). In

37

general, by Rieffel’s lemma [Ri, 1974], Lemma 2.4, C?h(G,A;σ) ∼= C?

r (G,A;σ).

In fact, equip L2(G,A) with the A-valued inner product:

〈ξ, η〉A =

∫G

ξ(g)?η(g)dg.

By a straightforward computation, 〈ϕ ?σ ξ, η〉A = 〈ξ, ϕ? ?σ η〉A for ϕ ∈

L∞c (G,A), so all elements of L∞c (G,A) have bounded adjoints. Hence, L2(G,A)

is a Hilbert C?h(G,A;σ)-module. By [Ri, 1974], Lemma 2.4, we have C?

h(G,A;σ) ∼=

C?r (G,A;σ).

Definition 3.2. Let A be a C?-algebra upon which the group G acts by α.

Let BG(A) = those kernel functions k : G×G→ A with k ∈ L∞(G×G,A)

(the set of all L∞ functions on G×G with values in A), such that the support

of k is contained in a bounded neighborhood of the diagonal of G×G. The

last condition amounts to saying that k(x, y) = 0 if xy−1 /∈ K, for some

compact subset K of G, which depends on k.

We make BG(A) into a ?-algebra with multiplication

k ?σ l(x, y) =

∫G

k(x, z)αxz−1(l(z, y))σ(xz−1, zy−1)∆(z)−1dz,

and ?-operation

(k)?(x, y) = αxy−1(k(y, x)?)∆(yx−1).

38

Note first that the cocycle conditions imply the following identities: σ(x−1, xy−1) =

σ(x, y−1), and σ(x, y−1)σ(y, x−1) = 1. Note also that in the integral expres-

sion for multiplication, the integrand is zero unless xz−1 and zy−1 both lie

in some compact set K depending on k and l. Hence for each x and y, z is

restricted to a compact subset of G and the integral makes sense. Also, the

integral is nonzero only if xy−1 ∈ K2, so k ?σ l ∈ BG(A).

We define the twisted kernel operator kα on L2(G,A) associated with the

kernel k ∈ BG(A) by

kα ?σ ξ(x) =

∫G

αx−1(k(x, y))ξ(y)σ(x−1, xy−1)∆(y)−1dy.

One checks that under the map k → kα, the image of BG(A) is a ?-subalgebra

BαG(A) of bounded Hilbert A-module maps with involution. In fact k → kα

is a ?-algebra homomorphism, with respect to the A-valued inner product

〈·, ·〉A on L2(G,A) defined in (3.1). Also, kα is a bounded operator by the

following estimate, which uses the definition of kα ?σ ξ above.

‖kα ?σ ξ‖22 ≤

∫ (∫‖k(x, y)‖A‖∆−1ξ(y)‖Ady

)2

dx ≤M

∫ (∫χK(xy−1)‖∆−1ξ(y)‖Ady

)2

dx

= M

∫ (∫χK(y)‖ξ(y−1x)‖Ady

)2

dx ≤Mµ(K)2‖ξ‖22,

where χK denotes the characteristic function of the set K, which is compact

39

and so has finite measure.

We would like to call this algebra, BαG(A), an (α, σ)-twisted Roe algebra

with coefficients in A. The standard Roe algebras were constructed (using

continuous functions instead of L∞-functions and without twisting nor co-

efficients) and studied by J. Roe in his study of index theory of geometric

operators on an open manifold. Elements in Roe algebras correspond to

kernel operators of finite propagation speed [Roe, 1993].

The closure of BαG(A) is denoted by Bα

G(A). There is also a natural right

G-action on BαG(A) given by the diagonal action βg(k)(x, y) = k(xg, yg)

on BαG(A). Let Bα

G(A)β be the G-invariant part of BαG(A). An element

k ∈ BαG(A)β satisfies k(xg, yg) = k(x, y) for all g ∈ G. This, in particular,

implies that k(xy−1, e) = k(x, y). We set k(x) = k(x, e) for k ∈ BαG(A)β.

Lemma 3.3. β gives an action of the group G on the space BαG(A), which

is isometric for the operator norm on BαG(A). The map g 7→ βg is a homo-

morphism from the group G into the group of ?-automorphisms of BαG(A).

Proof: In order to show that β is isometric, it suffices to show that k and

βx(k) have the same norms as operators on L2(G,A). For ξ, η ∈ L2(G,A), set

ξx(z) = αx−1(ξ(zx−1))σ(x, z−1)∆(x)−1/2 and ηx(y) = αx−1(η(yx−1))σ(x, y−1)∆(x)−1/2.

It is easy to check that 〈ξx, ξx〉 = αx−1(〈ξ, ξ〉), and similarly for η. Hence

40

ξx, ηx have the same norms as ξ, η, respectively, in L2(G,A). Noticing that

σ(x, y−1)σ(xy−1, yz−1) = σ(x, z−1)σ(y−1, yz−1), we have

〈 βx(k) ?σ ξ, η〉 =

∫(βx(k) ?σ ξ)(y)?η(y)dy

=

∫ ∫ (αy−1(βx(k)(y, z))∆−1ξ(z)σ(y−1, yz−1)

)?η(y)dzdy

=

∫ ∫ (αxy−1(k(y, z))∆−1ξ(zx−1)σ(xy−1, yz−1)

)?η(yx−1)d(zx−1)d(yx−1)

= αx

(∫ ∫ (αy−1(k(y, z))∆−1ξx(z)σ(y−1, yz−1)

)?ηx(y)dzdy

)= αx(〈k ?σ ξx, ηx〉),

and so βx(k) has the same norm as k as an operator on L2(G,A). To prove

the second statement, we need only to show that, for each g ∈ G, βg is a

?-automorphism. In fact,

βg(kα ?σ l

α)(x, y) =

∫G

k(xg, z)αxgz−1(l(z, yg))σ(xgz−1, zg−1y−1)∆(z)−1dz

=

∫G

k(xg, ug)αxg(ug)−1(l(ug, yg))σ(xu−1, uy−1)∆(ug)−1d(ug)

=

∫G

k(xg, ug)αxu−1(l(ug, yg))σ(xu−1, uy−1)∆(u)−1du

= βg(k)α ?σ βg(l)α(x, y),

and

βg((kα)?)(x, y) = (kα)?(xg, yg) = αxy−1(k(yg, xg)?)∆(yx−1) = (βg(k)α)?(x, y).

�

41

Proposition 3.4. Let β be as in Lemma 3.3. Then the fixed point algebra

BαG(A)β is naturally isomorphic to C?

h(G,A;σ) ∼= C?r (G,A;σ).

Proof: It is clear that BαG(A)β ∼= L∞c (G,A) in a natural way. In fact, if ϕ ∈

L∞c (G,A), then kϕ(x, y) = ϕ(xy−1) is obviously β-invariant and contained

in L∞(G × G,A). If ϕ has compact support K, kϕ(x, y) = 0 if xy−1 /∈ K,

so kϕ ∈ BG(A). On the other hand, given k ∈ BαG(A)β, let k ∈ L∞c (G,A)

be defined as k(y) = k(y, e). Since k(y, e) = 0 if y /∈ Kk, k has compact

support, and so lies in L∞c (G,A). The correspondences k 7→ k and ϕ 7→ kϕ

thus give a bijection BαG(A)β ∼= L∞c (G,A), which is easily seen to be a ?-

algebra isomorphism.

By invariance, k(x, y) = k(xy−1, e), for all x, y ∈ G. Hence, for ξ ∈

L2(G,A), k ∈ BαG(A)β, we have

kα ?σ ξ(x) =

∫G

αx−1(k(x, y))∆−1ξ(y)σ(x−1, xy−1)dy

=

∫G

αx−1(k(xy−1, e))∆−1ξ(y)σ(x−1, xy−1)dy

=

∫G

αx−1(k(y))ξ(y−1x)σ(x−1, y)dy = k ?σ ξ(x),

so ‖k‖BαG(A)β

= ‖k‖C?h(G,A,σ), since the topologies on both C?-algebras are

given by the operator norm on L2(G,A). This implies that BαG(A)β ∼=

C?h(G,A;σ). �

42

Lemma 3.5. For k ∈ BαG(A), define γt(k)(x, y) = exp(it(`(x)−`(y)))·k(x, y)

for t ∈ R. Then γt extends to a strongly continuous action of R on the C?-

algebra BαG(A).

Proof: For k ∈ BαG(A), ξ, η ∈ L2(G,A), we have 〈γt(k)?σξ, η〉A = 〈k?σ ξ, η〉A,

where ξ(y) = exp(−it`(y))ξ(y) and η(x) = exp(−it`(x))η(x). Also, ξ, η

have the same norms as ξ, η respectively in L2(G,A), and so γ is norm

preserving onBαG(A). It is also straightforward to check that for k, l ∈ Bα

G(A),

γs+t(k) = γs(γt(k)), γt(k?) = γt(k)?, and γt(k ?σ l) = γt(k) ?σ γt(l).

It remains to show the strong continuity. For k ∈ BαG(A), k(x, y) = 0 if

xy−1 /∈ K for some compact subset K of G. Therefore, for such x, y in the

support, |`(x) − `(y)| ≤ `(xy−1) ≤ C, since ` is bounded on compact sets.

Therefore, for t, s ∈ R such that |s− t| ≤ ε, we have

| exp(it(`(x)− `(y)))− exp(is(`(x)− `(y)))| ≤ (εC)/(2π).

Hence, for ξ ∈ L2(G,A),

‖ γt(kα) ?σ ξ(x)− γs(kα) ?σ ξ(x)‖2

2

=

∫G

∥∥∥∥∫G

(eit(`(x)−`(y)) − eis(`(x)−`(y)))αx−1(k(x, y))∆−1ξ(y)σ(x−1, xy−1)dy

∥∥∥∥2

A

dx

≤ (εC)

2π

∫G

(∫xy−1∈K

‖k(x, y)‖A‖∆−1ξ(y)‖Ady)2

dx ≤ (εC)

2π(‖|k|‖BG(C))

2‖|ξ|‖22,

where |k|(x, y) = ‖k(x, y)‖A, |ξ|(x) = ‖ξ(x)‖A. Clearly, |k| ∈ BG(C), and

43

|ξ| ∈ L2(G). The lemma is proved. �

We would like the C∞-vectors for the action of γ on BαG(A) to be equal to

S`2(G,A;σ), when intersected with C?h(G,A;σ). However, we can only prove

this if G is discrete (see Proposition 3.10 below). (For a counterexample,

consider the circle group (Remark 1.11).) We will therefore replace BαG(A)

with an appropriate dense ideal in order to make things work out. We begin

with a definition.

Definition 3.6. We denote by L2,∞∆ (G,A) the Banach space L∞(G,L2(G,A),∆−1/2),

the set of L∞-functions from G to L2(G,A), scaled against ∆−1/2. We define

the norm ‖ · ‖2,∞ on ϕ ∈ L2,∞∆ (G,A) by

‖ϕ‖2,∞ = esssup{‖ϕ(y)‖2∆(y)−1/2 | y ∈ G}.

Theorem 3.7. BαG(A) acts on L2,∞

∆ (G,A) as a left Banach module. Proof:

Let ξ ∈ L2,∞∆ (G,A). We assume that ξ is given by a function ξ(x, y) on

G×G, such that for each each y ∈ G, ξ(·, y) is in L2(G,A). Let k ∈ BαG(A).

We define

kα ?σ ξ(x, y) =

∫G

k(x, z)αxz−1(ξ(z, y))σ(xz−1, zy−1)∆(z)−1dz.

44

It is easy to verify the cocycle identity: σ(xz−1, zy−1) = σ(x, y−1)σ(y, z−1)σ(x−1, xz−1).

Hence,

(‖(kα ?σ ξ)(·, y)‖2)2 =

∫ ∥∥∥∥∫ k(x, z)αxz−1(ξ(z, y))σ(xz−1, zy−1)∆(z)−1dz

∥∥∥∥2

dx

=

∫ ∥∥∥∥∫ αx−1(k(x, z))αz−1(ξ(z, y))σ(x, y−1)σ(y, z−1)σ(x−1, xz−1)∆(z)−1dz

∥∥∥∥2

dx

=

∫ ∥∥∥∥∫ αx−1(k(x, z))αz−1(ξ(z, y))σ(y, z−1)σ(x−1, xz−1)∆(z)−1dz

∥∥∥∥2

dx

= (‖kα ?σ ξ(·, y)‖2)2 ≤ (‖kα‖BαG(A))2(‖ξ(·, y)‖2)2,

where ξ(z, y) = αz−1(ξ(z, y))σ(y, z−1). Hence ‖kα ?σ ξ‖2,∞ = esssupy(‖(kα ?σ

ξ)(·, y)‖2∆(y)−1/2) ≤ ‖kα‖BαG(A) esssupy‖ξ(·, y)‖2∆(y)−1/2 = ‖kα‖BαG(A) ‖ξ‖2,∞,

so the action is continuous. ¿From the fact that the multiplication in BαG(A)

(Definition 3.2) is associative, it is easy to see that (kα?σlα)?σξ = kα?σ(lα?σξ)

for kα, lα ∈ BαG(A), ξ ∈ L2,∞

∆ (G,A). �

Definition 3.8. Note that BG(A) is actually contained in L2,∞∆ (G,A) as

functions on G×G. (This is the reason for the factor of ∆−1/2 in Definition

3.6.) Let J be the completion of BG(A) in the weakest topology which

dominates both the topology on BαG(A) and the topology on L2,∞

∆ (G,A). It

is obvious that R acts on the Banach space L2,∞∆ (G,A) by the same formula

as γ on BαG(A) (see Lemma 3.5). The action is also isometric and strongly

continuous on BG(A) in the topology induced from L2,∞∆ (G,A). Thus γ acts

45

isometrically and strongly continuously on the Banach space J .

Corollary 3.9. The Banach space J so defined is a dense left Banach ideal

of BαG(A). In fact, ‖kα ?σ ξ‖J ≤ ‖kα‖BαG(A)‖ξ‖J .

Proof: This follows from the BαG(A)-module structure on L2,∞

∆ (G,A) (see

proof of Theorem 3.7), and the fact that the action of BαG(A) on L2,∞

∆ (G,A)

coincides with the multiplication in BαG(A). �

Proposition 3.10. Suppose that the group (G, `) has property RD. Let J∞

denote the set of C∞-vectors for the action γ on J . Then

J∞ ∩BαG(A)β = S`2(G,A;σ).

If, moreover, G is discrete, then

BαG(A)β

∞= S`2(G,A;σ),

where BαG(A)β

∞denotes the the set of C∞-vectors of γ on Bα

G(A)β. If (G, `)

is not RD, the inclusions ⊆ still hold, but we do not have equality in general.

Remark. Note that C?h(G,A;σ) is identified with Bα

G(A)β in a natural way

as noted in (3.4). The isometry is given by ϕ 7−→ ϕ, where ϕ(x, y) = ϕ(xy−1).

Proof: Let δ be the derivation obtained from the action γ of R on J . Let

βg(k)(x, y) = k(xg, yg) be the action β of G on BαG(A) from Lemma 3.3. Let

46

k ∈ J∞ ∩ BαG(A)β. Then k(x, y) = k(xy−1), where k(x) = k(x, e). Since

k ∈ J∞,

∞ > ‖δp(k)‖2,∞ = esssupy∈G

(∫G

‖(`(x)− `(y))pk(xy−1)‖2Adx

)1/2

∆(y)−1/2.

For all positive integers p, it is possible to find a fixed y for which this is

finite. We have,

∞ >

∫G

‖(`(uy)− `(y))pk(u)‖2Adu.

By induction,

∞ >

∫G

`(uy)2p‖k(u)‖2Adu ≥

∫G

(`(u)− `(y))2p‖k(u)‖2Adu.

By induction again,

∞ >

∫G

`(u)2p‖k(u)‖2Adu.

Since this holds for all p, it follows that k ∈ S`2(G,A;σ). Note that this did

not use the fact that (G, `) is RD.

Conversely, assume that ϕ ∈ S`2(G,A;σ) and (G, `) has property RD. Let

ϕ(x, y) = ϕ(xy−1). Then

δp(ϕ)(x, y) = ip(`(x)− `(y))pϕ(x, y),

47

and, using the fact that G is unimodular (Theorem 2.2),

‖ δp(ϕ)‖2,∞ = esssupy

(∫G

(`(x)− `(y))2p‖ϕ(x, y)‖2Adx

)1/2

≤ esssupy

(∫G

`(xy−1)2p‖ϕ(xy−1)‖2Adx

)1/2

= esssupy

(∫G

`(u)2p‖ϕ(u)‖2Adu

)1/2

= ‖|ϕ|‖p.

Here ‖|ϕ|‖p is the seminorm on S`2(G,A) defined in (1.2). Therefore, δpϕ ∈

L2,∞(G,A). Also, if ξ ∈ L2(G,A), then by the proof of Lemma 3.5, we have

‖(δpϕ) ?σ ξ‖22 ≤

∫G

∥∥∥∥∫G

(`(x)− `(y))pαx−1(ϕ(xy−1)σ(x−1, xy−1)ξ(y)dy

∥∥∥∥2

A

dx

≤∫G

∥∥∥∥∫G

(`(x)− `(s−1x))pαx−1(ϕ(s))σ(x−1, s)ξ(s−1x)ds

∥∥∥∥2

A

dx

≤ ‖ϕp‖2C?r (G)‖ξ‖2

2 ≤ c2|‖ϕ‖|2d‖ξ‖22,

for some constants d > 0, c > 0, by Propostion 2.3, where once again ϕp(x) =

`(s)p‖ϕ(s)‖A. Hence ϕ ∈ J∞ ∩ C?h(G,A).

If G is discrete, then G is also unimodular, and for any k ∈ BαG(A)β

∞, we

have

∞ > ‖δp(k)‖ = sup{‖ δp(k) ?σ ξ‖2 | ξ ∈ L2(G,A), ‖ξ‖2 = 1}

= sup‖ξ‖2=1

(∫G

∥∥∥∥∫G

(`(x)− `(y))pαx−1(k(x, y))ξ(y)σ(x−1, xy−1)dy

∥∥∥∥2

A

dx

)1/2

≥(∫

G

‖`(x)pαx−1(k(x, e))‖2Adx

)1/2

=

(∫G

`(x)2p‖k(x)‖2Adx

)1/2

= ‖|k|‖p,

by evaluating at ξ = δe, the step function at e = 1G. It follows that k ∈

48

S`2(G,A;σ). The converse follows from the same argument as for a general

group G above. This completes the proof of Proposition 3.10. �

Lemma 3.11. Let I be a dense left Banach ideal in a Banach algebra B.

Moreover, assume that ‖bϕ‖I ≤ ‖b‖B‖ϕ‖I for b ∈ B, ϕ ∈ I. Assume that

R acts strongly continuously on I, and let I∞ denote the Frechet algebra of

C∞-vectors. Then I∞ satisfies the DSC in B.

Proof: Let δ be the infinitesimal generator for the action of R on I. Let

ϕ, ψ ∈ I. Topologize I∞ by the seminorms

‖ϕ‖0 = ‖ϕ‖B,

‖ϕ‖k =1

k!‖δkϕ‖B +

1

(k − 1)!‖δk−1ϕ‖I , k = 1, 2, 3, . . . .

We have

‖ϕ? ψ ‖k ≤∑i+j=k

1

i!‖δiϕ‖B

1

j!‖δjψ‖B +

∑i+j=k−1

1

i!‖δiϕ‖B

1

j!‖δjψ‖I

=∑i+j=k

1

i!‖δiϕ‖B

1

j!‖δjψ‖B +

∑i+j=k,j 6=0

1

i!‖δiϕ‖B

1

(j − 1)!‖δj−1ψ‖I

≤ 1

k!‖δkϕ‖B‖ψ‖0 +

∑i+j=k,j 6=0

1

i!‖δiϕ‖B

(1

j!‖δjψ‖B +

1

(j − 1)!‖δj−1ψ‖I

)=

∑i+j=k

1

i!‖δiϕ‖B‖ψ‖j ≤

∑i+j=k

‖ϕ‖i‖ψ‖j.

This proves the Lemma. �

49

Proof of Theorem 2.5: By Lemma 3.11 and Corollary 3.9, J∞ satisfies

the DSC in BαG(A). By Proposition 3.10, J∞ ∩ C?

h(G,A;σ) = S`2(G,A;σ) if

(G, `) is RD. Hence the seminorms on J∞ and the norm on BαG(A) agree with

the seminorms on S`2(G,A;σ) and the norm on C?h(G,A;σ) respectively. So

S`2(G,A;σ) satisfies the DSC in C?h(G,A;σ).

If G is discrete, BαG(A)

∞∩ C?

h(G,A;σ) = S`2(G,A;σ) (Proposition 3.10)

implies more directly that S`2(G,A;σ) satisfies DSC in BαG(A), and hence in

C?h(G,A;σ). �

4 References

[BC, 1988] P. Baum and A. Connes, Chern character for discrete groups, Collection: A

fete of topology., Academic Press, Orlando, Florida, (1988), 163-232.

[BCH, to appear] P. Baum, A. Connes and N. Higson, Classifying space for proper

actions and K-theory of group C?-algebras, Contempory Mathematics, Amer. Math. Soc,

to appear.

[Bl, 1986] B. Blackadar, K-theory for Operator Algebras, MSRI Publ., 5, Springer-

Verlag, 1986.

[BlCu, to appear] B. Blackadar and J. Cuntz, Differential Banach algebra norms and

50

smooth subalgebras of C*-algebras, J. Operator Theory to appear.

[Bo, 1990] J. Bost, Principe d’Oka and K-therie et systeme dynamiques non-commutatifs,

Invent. Math. 101 (1990), 261-333.

[CaPh, 1991] A. Carey and J. Phillips, Algebras almost commuting with Clifford algebras

in a II∞ factor, K-Theory 4 (1991), 445-478.

[Co, 1985] A. Connes, Non-commutative differential geometry, Publ. Math. I.H.E.S. 62

(1985), 41-144.

[CM, 1990] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and

hyperbolic groups, Topology 29 (1990), 345-388.

[Gr, 1981] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math

I.H.E.S. 53 (1981). 53-73.

[Gr, 1987] M. Gromov, Hyperbolic groups, Essays in Group Theory, MSRI Publ.,

Springer-Verlag 8 (1987), 75-263.

[PHa, 1988] P. de la Harpe, Groupes hyperboliques, algebres d’operateures et un theoreme

de Jolissaint, C. R. Acad. Sci. Paris 307, Serie I (1988), 771-774.

[HR, 1979] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer-Verlag.

Berlin-Heidelberg-New York, 1979.

[Je, 1973] J. W. Jenkins, Nonsymmetric groups algebras, Studia Math. 45 (1973),

295-307.

[Ji, 1992] R. Ji, Smooth dense subalgebras of reduced group C*-algebras, Schwartz coho-

mology of groups, and cyclic cohomology, J. Funct. Anal. 107 (1992), 1-33.

[Ji, 1993] R. Ji, A module structure on cyclic cohomology of group graded algebras,

K-theory 7 (1993), 369-399.

51

[Jo, 1990] P. Jolissaint, Rapidly deceasing functions in reduced group C*-algebras of

groups, Trans. Amer. Math. Soc. 317 (1990), 167-196.

[Jo, 1989] P. Jolissaint, K-theory of reduced C*-algebras and rapidly deceasing functions

on groups, K-theory 2(6) (1989), 723-736.

[JoVa, 1991] P. Jolissaint and A. Valette, Normes de Sobolev et convoluteurs bornes sur

L2(G), Ann. L’Inst. Fourier Tome 41 (1991), 797-822.

[Pa, 1978] T. W. Palmer, Classes of nonabelian, noncompact locally compact groups,

Rocky Mountain J. Math. 8 (1978), 683-741.

[Ped, 1979] G. Pedersen, C*-algebras and Their Automorphism Groups, Academic Press,

London/New York/San Francisco, 1979.

[Prom, 1985] D. Promislow, Equivalence classes of length functions on groups, Proc.

London Math. Soc. (3)51 (1985), 449-477.

[Ri, 1974] M. A. Rieffel, Induced representations of C?-algebras, Adv. Math. 13 (1974),

176-257.

[Roe, 1993] J. Roe, Coarse cohomology and index theory for complete Riemannian

manifolds, Mem. Amer. Math. Soc., No. 497, 1993.

[Sch, 1992] L. B. Schweitzer, A short proof that Mn(A) is local if A is local and Frechet,

Internat. J. Math. 3(4) (1992), 581-589.

[Sch, 1993] L. B. Schweitzer, Summary of spectral invariance results, C.R. Math. Acad.

Sciences Canada, XV(1) (February, 1993), 13-18.

[Sch, 1993a] L. B. Schweitzer, Spectral invariance of dense subalgebras of operator alge-

bras, Internat. J. Math. 4(2) (1993), 289-317.

[Sch, 1993b] L. B. Schweitzer, Dense m-convex Frechet subalgebras of operator algebra

52

crossed products by Lie groups, Internat. J. Math. 4(4) (1993), 601-673.

[Tr, 1967] F. Treves, Topological Vector Spaces, Distributions, and Kernels, Academic

Press, New York, 1967.

[Vi, 1990] MF Vigneras, On formal dimensions for reductive p-adic groups, Israel Math.

Conf. Proc. 2 (1990), 255-266.

[Vi, 1992] MF Vigneras, Homologie cyclique, Principe de Selberg et pseudo-coefficient,

preliminary version, 1992.

[Wa, 1971] L. Waelbroeck, Topological Vector Spaces and Algebras, Lect. Notes in Math.

230, Springer-Verlag, 1971.

[ZM, 1968] G. Zeller-Meier, Produits croises d’une C*-algebre par un groupe d’automorphismes,

J. Math. Pures Appl. 47 (1968), 101-239.

Web Page: http://www.svpal.org/~lsch/Math/indexMath.html.

53