amenability for fell bundles

31 Pages·2008·0.25 MB·English

AMENABILITY FOR FELL BUNDLES
Ruy Exel*
Departamento de Matem´atica
Universidade de S˜ao Paulo
Rua do Mat˜ao, 1010
05508-900 S˜ao Paulo – Brazil
[email protected]
FAX [phone]
Abstract. Given a Fell bundle B, over a discrete group Γ, we con-
struct its reduced cross sectional algebra C∗(B), in analogy with the
r
reduced crossed products defined for C∗-dynamical systems. When the
reduced and full cross sectional algebras of B are isomorphic, we say that
the bundle is amenable. We then formulate an approximation property
which we prove to be a sufficient condition for amenability.
A theory of Γ-graded C∗-algebras possessing a conditional expec-
tation is developed, with an eye on the Fell bundle that one naturally
associates to the grading. We show, for instance, that all such algebras
are isomorphic to C∗(B), when the bundle is amenable.
r
We also study induced ideals in graded C∗-algebras and obtain a
generalization of results of Strˇatilˇa and Voiculescu on AF-algebras, and
of Nica on quasi-lattice ordered groups. A brief comment is made on
the relevance, to our theory, of a certain open problem in the theory of
exact C∗-algebras.
An application is given to the case of an F –grading of the Cuntz–
n
Krieger algebras O , recently discovered by Quigg and Raeburn. Specif-
A
ically, we show that the Cuntz–Krieger bundle satisfies the approxima-
tion property, and hence is amenable, for all matrices A with entries in
{0,1}, even if A does not satisfy the well known property (I) studied by
Cuntz and Krieger in their paper.
1. Introduction. A possible definition of Fell bundles (also called C∗-algebraic bundles
[14]), for the special case of discrete groups, states that these are given by a collection B =
(cid:0) (cid:1)
B of closed subspaces of a C∗-algebra B, indexed by a discrete group Γ, satisfying
t t∈Γ
B∗ = B and B B ⊆ B for all t and s in Γ. If, in addition, the B ’s are linearly
t t−1 t s ts t
independent and their direct sum is dense in B, then B is said to be a graded C∗-algebra.
Fell bundles and graded algebras occur in an increasing number of situations in the
theory of C∗-algebras, often without the perception that they are there. The example in
which the Fell bundle structure is the most conspicuous, is that of the well known crossed
*
Partially supported by CNPq, Brazil.
amenability for fell bundles 2
product construction associated to a C∗-dynamical system [18], recently extended to the
case of twisted partial dynamical systems [11]. See also [9, 16].
In the most tractable cases, these bundles have a commutative unit fiber algebra,
that is, B (where e denotes the unit group element). In that case, by the main result of
e
[11], one can say that, up to stabilization, the complexity of the bundle resides in three
distinct compartments, namely, the topological structure of the spectrum of B , certain
e
homeomorphisms between open sets in that spectrum, and a certain two cocycle. See [11]
for more details.
Among the examples in which the Fell bundle structure is not so striking to the eye,
lie some of the most intensely studied C∗-algebras of the past couple of decades. These
include all the AF-algebras [4,22,10], the Cuntz–Krieger algebras O [5,20], algebras
A
generated by Wiener-Hopf operators [17], non-commutative Heisenberg manifolds [21,1],
the quantum SU groups [24](here one mayuse one ofseveraleasily available circle actions
2
to obtain many interesting gradings, as done in [9]), the soft tori A [7, 8, 6], and many
ε
others.
However intriguing this may be, most of the examples cited possess a commutative
unit fiber algebra, and hence the comment above applies. But, there is a catch. The bundle
structure alone is not enough to characterize the algebra. The point is that non-isomorphic
graded algebras may possess identical associated Fell bundles, as in the case of the reduced
and full group C∗-algebras of non-amenable discrete groups (see [14, VIII.16.12]).
Itisthemainpurposeofthisworktostudythis pointindetail. Thecruxofthematter
is thus to determine conditions on a Fell bundle B, such that all graded C∗-algebras, whose
associated Fell bundle coincides with B, are isomorphic to each other. After we show that
all such algebras lie in between the full and reduced cross sectional algebras of the bundle,
that is C∗(B) and C∗(B), respectively, this is equivalent to saying that the left regular
r
representation of C∗(B) is faithful.
Inspired by the work of Andu Nica [17], we call such bundles amenable. Our main
contribution is to formulate an approximation property for Fell bundles, which we prove
to be a sufficient condition for amenability. This condition is strongly influenced by the
work of Claire Anantharaman-Delaroche and, to a certain extent, could be thought of as
an attempt at a generalization of a similar condition studied in [2]. We do not claim to
have taken the analogy to its limits, as the role of the center of the unit fiber algebra,
played in [2], is yet to be understood in the Fell bundle situation.
Our major application is to the case of the recently discovered bundle structure, over
the free group F , of the Cuntz–Krieger algebras O , obtained by Quigg and Raeburn
n A
in [20], in terms of a co-action of F . To study this example, we show that the Cuntz–
n
Krieger relations, that is, the relations that define the algebras O , give rise to a partial
A
representation [13] of the free group. By a partial representation of a group Γ on a
Hilbert space H, we mean a unital map σ:Γ → L(H), such that, for all t,r ∈ Γ, one has
σ(t−1) = σ(t)∗, and σ(t)σ(r)σ(r−1) = σ(tr)σ(r−1). See [13] for more details.
Inturn,givenanypartialrepresentationofadiscretegroup,weconstructanassociated
Fell bundle. This construction ascribes the Fell bundle related to O , mentioned above, to
A
amenability for fell bundles 3
the partial representation arising from any universal representation of the Cuntz–Krieger
relations.
Irrespective of the main hypothesis imposed on the matrix A in [5], namely, condition
(I), we show that the Cuntz–Krieger bundle satisfies the approximation property and hence
is amenable. In fact, we prove that this holds for the Fell bundle associated to any semi-
saturated (see below) partial representation σ of F , which satisfies
n
n
X
σ(g )σ(g )∗ = 1,
i i
i=1
where g ,...,g are the generators of the free group.
1 n
On another front, we study induced ideals in graded algebras, where we are able
to mimic Nica’s work in [17] and obtain results very similar to his. In fact, the work
of Nica has been an inspiration for us all along, as he also treats questions related to the
approximation property. Other than merely formal generalizations, we seem to have gotten
a little further, because our proof of the amenability of the Cuntz–Krieger bundle is given
independently of the work of Cuntz and Krieger, themselves, namely, the uniqueness of the
C∗-algebra generated by non-trivial representations of the Cuntz–Krieger relations, when
property (I) is present [5].
Still under the heading of induced ideals, we point out a curious relationship with
an open problem in the theory of exact C∗-algebras, stated in [23, 2.5.3]. The question
of whether C∗(Γ) is an exact C∗-algebra for any countable discrete group Γ, seems to be
r
related to what we do here.
The author is indebted to several people who contributed in many ways for the evolu-
tionofthiswork. AmongthesehewouldliketoexpresshisthankstoClaireAnantharaman-
Delaroche for a brief, but fruitful discussion during his short visit to Orleans, to Marcelo
Laca for pointing out important references and also for several interesting conversations,
and to Cristina Cerri for having carefully gone over the paper [2] at the operator algebra
seminar in S˜ao Paulo and for many discussions as well.
2. Reduced Cross Sectional Algebras. Throughout this section, Γ will denote a
discrete group and B = {B } will be a fixed C∗-algebraic bundle over Γ, as defined
t t∈G
in [14]. In recent years it has been customary to refer to C∗-algebraic bundles as Fell
bundles, a terminology we think is quite appropriate.
Our specialization to the case of discrete groups has the primary purpose of avoiding
technical details which occur in the theory of Fell bundles over continuous groups. With
some more work, we believe that much, if not all, that we do here can be extended to the
general case. Another reason for us being satisfied with the discrete group case is that our
main application is for Γ = F , the free group on n generators.
n
By a section of B we mean any function
[
f:Γ → B ,
t
t∈Γ
amenability for fell bundles 4
such that f(t) ∈ B for all t ∈ Γ.
t
Let C (B) denote the set of all finitely supported sections of B. We shall regard C (B)
c c
as a right module over B , which, when equipped with the B –valued inner product
e e
X
hξ,ηi = ξ(t)∗η(t), ξ,η ∈ C (B),
c
t∈Γ
becomes a pre-Hilbert module in the sense of [15, 1.1.1]. The completion of C (B) can be
c
shown to consist of all cross sections ξ of B such that the series
X
ξ(t)∗ξ(t)
t∈Γ
P
converges unconditionally. Incidentally, we say that a series x in a Banach space E,
i∈I i
indexed by any index set I, is unconditionally summable to x ∈ E if, for any ε > 0, there
exists a finite subset I ⊆ I such that, for all finite subsets J ⊆ I, with I ⊆ J, one has
0 0
X
kx− x k < ε.
i
i∈J
As we said before, we believe in the possibility of generalizing the results in this work
to Fell bundles over continuous group. If that is to be attempted, then the unconditional
summability, which is such a pervasive ingredient here, is likely to be replaced by the
concept of unconditional integrability of [12].
We denote by l (B) the completion of C (B), so that l (B) becomes a right Hilbert
2 c 2
B –module. Given ξ and η in l (B) one can show that
e 2
X
hξ,ηi = ξ(t)∗η(t),
t∈Γ
where the series is unconditionally summable, as described above.
As in [15, 1.1.7], we will denote by L (l (B)), or simply by L(l (B)), the C∗-algebra
B 2 2
e
of all adjointable operators on l (B).
2
For each t ∈ Γ and each b ∈ B (the subscript in b is not absolutely necessary but it
t t t
will be used to remind us that b belongs to the fiber B ), define
t t
Λ(b )ξ| = b ξ(t−1s), ξ ∈ l (B)), s ∈ Γ.
t s t 2
It is not difficult to show that Λ(b )ξ does belong to l (B), and that kΛ(b )ξk ≤
t 2 t
kb kkξk.
t
2.1. Proposition. Λ(b ) is an adjointable operator on l (B) and Λ(b )∗ = Λ(b∗).
t 2 t t
amenability for fell bundles 5
Proof. Given ξ,η ∈ l (B) we have
2
X X X
hξ,Λ(b )ηi = ξ(s)∗b η(t−1s) = ξ(ts)∗b η(s) = (b∗ξ(ts))∗η(s) = ...
t t t t
s∈Γ s∈Γ s∈Γ
Observe that b∗ ∈ B , hence Λ(b∗)ξ| = b∗ξ(ts). So, the above equals
t t−1 t s t
X
... = (Λ(b∗)ξ| )∗η(s) = hΛ(b∗)ξ,ηi. ut
t s t
s∈Γ
2.2. Proposition. The map
[
Λ: B → L(l (B))
t 2
t∈Γ
is a representation of B in the sense that for all t,s ∈ Γ (compare [14, VIII.9.1])
i) Λ is a continuous linear map from B to L(l (B)),
t 2
ii) for b ∈ B and c ∈ B one has Λ(b c ) = Λ(b )Λ(c ),
t t s s t s t s
iii) Λ(b∗) = Λ(b )∗.
t t
Proof. Since (i) is easy and (iii) is already demonstrated, let us prove (ii). Take ξ ∈ l (B),
2
then, since b c ∈ B ,
t s ts
Λ(b c )ξ| = b c ξ(s−1t−1r) = b (cid:0)Λ(c )ξ| (cid:1)= Λ(b )Λ(c )ξ| . ut
t s r t s t s t−1r t s r
From now on we shall refer to Λ as the left regular representation of B.
2.3. Definition. The reduced cross sectional algebra of B, denoted C∗(B), is the sub
r
C∗-algebra of L(l (B)) generated by the range of the left regular representation of B.
2
We recall that the (full) cross sectional algebra of B, denoted C∗(B) [14, VIII.17.2],
is the enveloping C∗-algebra of l (B). By the universal property (see [14, VIII.16.12]) of
1
C∗(B) one sees that there is a canonical epimorphism of C∗-algebras, which we call the
left regular representation of C∗(B) and, by abuse of language, still denote by Λ,
Λ:C∗(B) → C∗(B).
r
We would now like to develop the rudiments of a Fourier analysis for C∗(B), and, in
r
particular, to define Fourier coefficients for elements of C∗(B). For each t ∈ Γ let
r
j :B → l (B)
t t 2
be the “inclusion” map, given by
(cid:26)
b if s = t
j (b )| = t
t t s 0 if s 6= t
amenability for fell bundles 6
for b ∈ B and s ∈ Γ
t t
We shall regard each B as a right Hilbert B –module under the obvious right module
t e
structure and the B –valued inner product given by
e
hb ,c i = b∗c , b ,c ∈ B .
t t t t t t t
This said, one can easily prove that each j is an adjointable map and that for ξ ∈ l (B),
t 2
one has j∗(ξ) = ξ(t). It follows that j∗j is the identity map on B , and hence that j is
t t t t t
¯
an isometry. This allows us to identify B and its image B = j (B ) within l (B).
t t t t 2
Incidentally, this shows the very subtle fact that any adjointable map from l (B) into
2
¯
any Hilbert B –module, remains an adjointable map if restricted to B , because restricting
e t
is the same as composing with j .
t
2.4. Proposition. Let t,s ∈ Γ, b ∈ B and c ∈ B . Then Λ(b )j (c ) = j (b c ).
t t s s t s s ts t s
¯ ¯
Therefore Λ(b )B ⊆ B . In addition, the map
t s ts
¯ ¯
b ∈ B 7→ Λ(b )| ∈ L(B ,B )
t t t B¯ e t
e
is isometric.
Proof. We have
(cid:0) (cid:1)
Λ(b )j (c )| = b j (c )| = δ b c = j (b c )| ,
t s s r t s s t−1r t−1r,s t s ts t s r
where δ is the Kronecker symbol. This proves the first statement. With respect to
t−1r,s
the isometric property of the map above, all we are saying is that
sup{kb ak:a ∈ B ,kak ≤ 1} = kb k.
t e t
This follows from the fact that any approximate unit for B acts as an approximate unit
e
for B also [14, VIII.16.3]. ut
t
2.5. Corollary. For each t ∈ Γ and each b ∈ B , one has that kΛ(b )k = kb k and hence
t t t t
each B may be identified with its image in C∗(B).
t r
Proof. Follows immediately from (2.4). ut
The following is the crucial step in defining Fourier coefficients.
2.6. Proposition. For each x in C∗(B) and each t in Γ there exists a unique b in B
r t t
such that j∗xj (a) = b a for every a ∈ B . In addition we have kb k ≤ kxk.
t e t e t
Proof. Uniqueness follows from the fact that if b a = 0 for all a ∈ B , then b = 0. As
t e t
for the existence part, the easiest case, by far, is when B has a unit. In this case b is
e t
just j∗xj (1). If no unit is available suppose, to start with, that x = P Λ(b ), where
t e s∈Γ s
b ∈ B and b = 0 except for finitely many group elements s. We then have
s s s
X
j∗xj (a) = j∗Λ(b )j (a) = b a.
t e t s e t
s∈Γ
amenability for fell bundles 7
This says that j∗xj , viewed as an element of L(B ,B ), coincides with Λ(b )| , and hence
t e e t t Be
lies in the isometric copy of B within L(B ,B ) provided by (2.4). Now, since the set of
t e t
all x0s considered is dense in C∗(B), we have obtained the existence part for all x ∈ C∗(B).
r r
Next observe that
kb k = kΛ(b )| k = kj∗xj k ≤ kxk. ut
t t Be t e
2.7. Definition. For x in C∗(B) and t ∈ Γ the tth Fourier coefficient of x is the unique
r
element xˆ(t) ∈ B such that j∗xj (a) = xˆ(t)a for all a in B . The Fourier transform of x
t t e e
is the cross section of B defined by t 7→ xˆ(t).
Given the left regular representation of C∗(B)
Λ:C∗(B) → C∗(B),
r
we can easily define the Fourier coefficients for elements of C∗(B) as well, that is, if
y ∈ C∗(B) we put yˆ(t) = Λd(y)(t).
P
From the proof of (2.6) we see that, if x is the finite sum x = Λ(b ) with b ∈ B ,
t∈Γ t t t
then xˆ(t) = b . Also it may be worth insisting that (2.6) yields kxˆ(t)k ≤ kxk.
t
2.8. Proposition. For x in C∗(B), t,s ∈ Γ and b ∈ B one has
r t t
j∗xj (b ) = xˆ(st−1)b .
s t t t
P
Proof. By continuity it is enough to consider finite sums x = Λ(c ) as above. For
r∈Γ r
such an x we have
X X
j∗xj (b ) = j∗Λ(c )j (b ) = j∗j (c b ) = c b = xˆ(st−1)b .
s t t s r t t s rt r t st−1 t t
r∈Γ r∈Γ
We have used that j∗j = 0 when s 6= r, a fact that is easy to see. ut
s r
The Fourier coefficient xˆ(e) has special properties worth mentioning.
2.9. Proposition. The map E:C∗(B) → B given by E(x) = xˆ(e) is a positive, con-
r e
tractive conditional expectation.
Proof. We first note that we are tacitly identifying B and its sibling Λ(B ), as permitted
e e
by (2.5). We have already seen that E is a contractive map. If x = Λ(b ) with b ∈ B ,
e e e
then we saw that E(x) = b , so E is idempotent.
e
P
If x = Λ(b ) as before, then
t∈Γ t
x∗x = X Λ(b )∗Λ(b ) = X Λ(b∗b ) = X Λ(b∗b ) = XΛ(cid:0)Xb∗b (cid:1).
t s t s t tr t tr
t,s∈Γ t,s∈Γ t,r∈Γ r∈Γ t∈Γ
Now, since P b∗b is in B , we have that E(x∗x) = P b∗b ≥ 0, so E is positive.
t∈Γ t tr r t∈Γ t t
P
Finally, if x = Λ(b ) is a finite sum, and if a ∈ B , then
t∈Γ t e
(cid:0)X (cid:1)
E(ax) = E Λ(ab ) = ab = aE(x)
t e
t∈Γ
and similarly, E(xa) = E(x)a. ut
amenability for fell bundles 8
2.10. Proposition. For x ∈ C∗(B) one has that the sum
r
X
xˆ(t)∗xˆ(t)
t∈Γ
(cid:0) (cid:1)
is unconditionally convergent, and hence ξ = xˆ(t) represents an element of l (B).
x t∈Γ 2
Also, for any a in B we have xj (a) = ξ a.
e e x
P
Proof. Suppose that x = Λ(b ) is a finite sum, with b ∈ B . Then, obviously
t∈Γ t t t
P
ξ = j (b ) belongs to l (B) and we have
x t∈Γ t t 2
X X X
xj (a) = Λ(b )j (a) = j (b a) = j (b )a = ξ a.
e t e t t t t x
t∈Γ t∈Γ t∈Γ
Also
X
kξ k2 = k b∗b k = kE(x∗x)k ≤ kx∗xk = kxk2.
x t t
t∈Γ
Therefore the relation x 7→ ξ ∈ l (B) defines a bounded map, which so far is defined only
x 2
for the x0s as above, but which may be extended to the whole of C∗(B) by continuity.
r
Again by continuity we have xj (a) = ξ a, for any x in C∗(B) and any a in B . Next
e x r e
observe that, for t ∈ Γ,
j∗xj (a) = j∗(ξ a) = j∗(ξ )a.
t e t x t x
(cid:0) (cid:1)
which implies, by (2.6), that j∗(ξ ) = xˆ(t). This says that ξ = xˆ(t) and the proof is
t x x t∈Γ
therefore concluded. ut
2.11. Corollary. For x ∈ C∗(B) one has
r
X
E(x∗x) = xˆ(t)∗xˆ(t).
t∈Γ
Proof. For a,b ∈ B we have
e
a∗E(x∗x)b = ha,xd∗x(e)bi = ha,je∗x∗xje(b)i = hxje(a),xje(b)i =
X
= hξ a,ξ bi = a∗hξ ,ξ ib = a∗ xˆ(t)∗xˆ(t)b,
x x x x
t∈Γ
from which the conclusion follows. ut
With this we arrive at an important result, which says, basically that E is a faithful
conditional expectation on C∗(B).
r
2.12. Proposition. For x ∈ C∗(B) the following are equivalent
r
i) E(x∗x) = 0
ii) xˆ(t) = 0 for every t ∈ Γ
iii) x = 0.
amenability for fell bundles 9
Proof. The equivalence of (i) and (ii) is a consequence of (2.11). That (iii) implies (ii) is
obvious, so let us prove that (ii) implies (iii). Assume xˆ(t) = 0 for all t ∈ Γ. Then, by
(2.8), it follows that j∗xj = 0 for all t and s in Γ.
s t
Now, note that any ξ ∈ l (B) is the sum of the unconditionally convergent series
2
ξ = P j j∗(ξ). So
t∈Γ t t
XX
xξ = j j∗xj j∗ξ = 0,
s s t t
s∈Γt∈Γ
that is, x = 0. ut
3. Graded C∗-algebras. In this section we will study the relationship between graded
C∗-algebrasandFellbundles. Gradedalgebrasoccurinagreatnumberofdifferentcontexts
in the theory of operator algebras, as in the theory of co-actions of discrete groups [19].
See also [11], [17], [13].
The following concept is taken from [14, VIII.16.11].
(cid:0) (cid:1)
3.1. Definition. Let B be a C∗algebra, Γ be a discrete group and let B be a
t t∈Γ
(cid:0) (cid:1)
collection of closed linear subspaces of B. We say that B is a grading for B if, for
t t∈Γ
each t,s in Γ one has
i) B∗ = B
t t−1
ii) B B ⊆ B
t s ts
L
iii) The subspaces B are independent and B is the closure of the direct sum B .
t t∈Γ t
In that case we say that B is a Γ–graded C∗-algebra. Each B is called a grading subspace.
t
The primary example of graded C∗-algebras is offered by the theory of Fell bundles.
3.2. Proposition. Let B be a Fell bundle over the discrete group Γ. Then C∗(B) is a
r
graded C∗-algebra via the fibers B of B.
t
Proof. The only slightly non trivial axiom to be proved regards the independence of the
P
B ’s. Assume, for that purpose, that x = Λ(b ) is a finite sum with b ∈ B and that
t t∈Γ t t t
x = 0. Then b = xˆ(t) = 0. ut
t
A similar reasoning shows that the full cross sectional algebra is also naturally graded.
Conversely, suppose we are given a graded C∗-algebra B = L B (by this notation we
t∈Γ t
wish to say that all of the conditions of (3.1) are verified). One can then construct a
Fell bundle over Γ, by taking the fibers of the bundle to be the grading subspaces. The
multiplication and adjoint operations, required on a Fell bundle, are defined by restricting
the corresponding operations on B.
In fact there is a great similarity between the formal definitions of Fell bundles and
that of a graded C∗-algebras. However, there are important conceptual differences, better
illustrated by the example provided by the full and reduced group C∗-algebras of a non-
amenable discrete group. These are non-isomorphic graded C∗-algebras whose associated
Fell bundles are indistinguishable from each other. See also [14, VIII.16.11].
amenability for fell bundles 10
Suppose we are given a Γ–graded C∗-algebra B = L B . Let us denote by B its
t∈Γ t
associated Fell bundle. Then, by the universal property of C∗(B) [14, VIII.16.11], there is
a unique C∗-algebra epimorphism
Φ:C∗(B) → B,
which is the identity on each B (identified both with a subspace of C∗(B) and of B in the
t
natural way). This says that C∗(B) is, in a sense, the biggest C∗-algebra whose associated
Fell bundle is B. Our next result will show that the reduced cross sectional algebra is on
the other extreme of the range. It is also a very economical way to show a C∗-algebra to
be graded.
3.3. Theorem. Let B be a C∗-algebra and assume that for each t in a discrete group
Γ, there is associated a closed linear subspace B ⊆ B such that, for all t and s in Γ one
t
has
i) B B ⊂ B ,
t s ts
ii) B∗ = B ,
t t−1
S
iii) the linear span of B is dense in B.
t∈Γ t
Assume, in addition, that there is a bounded linear map
F:B → B ,
e
such that F is the identity map on B and that F vanishes on each B , for t 6= e. Then
e t
(cid:0) (cid:1)
(a) The subspaces B are independent and hence B is a grading for B.
t t t∈Γ
(b) F is a positive, contractive, conditional expectation.
(c) If B denotes the associated Fell bundle, then there exists a C∗-algebra epimorphism
λ:B → C∗(B),
r
such that λ(b ) = Λ(b ) for each t in Γ and each b in B .
t t t t
P
Proof. If x = b is a finite sum with b ∈ B , then
t∈Γ t t t
x∗x = X b∗b = X(cid:0)Xb∗b (cid:1),
t s t tr
t,s∈Γ r∈Γ t∈Γ
so F(x∗x) = P b∗b .
t∈Γ t t
Therefore, if x = 0 then each b = 0, which shows the independence of the B ’s,
t t
and also that F is positive. Given a in B , it is easy to see that F(ax) = aF(x) and
e
F(xa) = F(x)a. So, apart from contractivity, (b) is proven.
Define a pre right Hilbert B –module structure on B via the B –valued inner product
e e
hx,yi = F(x∗y), x,y ∈ B.