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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). �
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