Author | Kenneth R. Davidson | |

Isbn | 9780821853023 | |

File size | 953.95KB | |

Year | 2011 | |

Pages | 53 | |

Language | English | |

File format | ||

Category | mathematics |

Number 982
Operator Algebras
for Multivariable Dynamics
Kenneth R. Davidson
Elias G. Katsoulis
January 2011 • Volume 209 • Number 982 (first of 5 numbers)
•
ISSN 0065-9266
Library of Congress Cataloging-in-Publication Data
Davidson, Kenneth R.
Operator algebras for multivariable dynamics / Kenneth R. Davidson, Elias G. Katsoulis.
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1. Operator algebras. 2. Multivariate analysis. 3. Dynamics. I. Katsoulis, Elias G., 1963II. Title.
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The second author dedicates this work to the memory of his father,
George E. Katsoulis
Contents
Chapter 1. Introduction
1.1. The one variable case
1.2. Universal operator algebras
1
3
4
Chapter 2. Dilation Theory
2.1. Dilation for the tensor algebra
2.2. Boundary representations and the C*-envelope
2.3. C*-correspondences
2.4. Dilation and the semi-crossed product
7
7
9
13
17
Chapter 3. Recovering the Dynamics
3.1. Fourier series and automatic continuity
3.2. Characters and nest representations
3.3. Piecewise conjugate multisystems
3.4. The main theorem
3.5. The converse
23
23
26
30
32
35
Chapter 4. Semisimplicity
4.1. Wandering sets and recursion
4.2. Semisimplicity
43
43
44
Chapter 5. Open Problems and Future Directions
49
Bibliography
51
v
Abstract
Let X be a locally compact Hausdorﬀ space with n proper continuous self
maps σi : X → X for 1 ≤ i ≤ n. To this we associate two conjugacy operator
algebras which emerge as the natural candidates for the universal algebra of the
system, the tensor algebra A(X, τ ) and the semicrossed product C0 (X) ×τ F+
n.
We develop the necessary dilation theory for both models. In particular, we
exhibit an explicit family of boundary representations which determine the C*envelope of the tensor algebra.
We introduce a new concept of conjugacy for multidimensional systems, called
piecewise conjugacy. We prove that the piecewise conjugacy class of the system
can be recovered from the algebraic structure of either A(X, σ) or C0 (X) ×σ F+
n.
Various classiﬁcation results follow as a consequence. For example, if n = 2 or
3, or the space X has covering dimension at most 1, then the tensor algebras are
algebraically isomorphic (or completely isometrically isomorphic) if and only if the
systems are piecewise topologically conjugate.
We deﬁne a generalized notion of wandering sets and recurrence. Using this,
it is shown that A(X, σ) or C0 (X) ×σ F+
n is semisimple if and only if there are no
generalized wandering sets. In the metrizable case, this is equivalent to each σi
being surjective and v-recurrent points being dense for each v ∈ F+
n.
Received by the editor September 14, 2007.
Article electronically published on June 8, 2010; S 0065-9266(10)00615-0.
2000 Mathematics Subject Classiﬁcation. Primary 47L55; Secondary 47L40, 46L05, 37B20,
37B99.
Key words and phrases. multivariable dynamical system, operator algebra, tensor algebra,
semi-crossed product, Cuntz-Pimsner C*-algebra, semisimple, radical, piecewise conjugacy, wandering sets, recurrence.
The ﬁrst author was partially supported by an NSERC grant.
Aﬃliation at time of publication: Kenneth R. Davidson, Pure Mathematics Department,
University of Waterloo, Waterloo, Ontario N2L–3G1, Canada; email: [email protected]
The second author was partially supported by a grant from ECU.
Aﬃliation at time of publication: Elias G. Katsoulis, Department of Mathematics, East
Carolina University, Greenville, North Carolina 27858; email: [email protected]; or Department
of Mathematics, University of Athens, 15784, Athens, Greece.
c
2010
American Mathematical Society
vii
CHAPTER 1
Introduction
Let X be a locally compact Hausdorﬀ space; and suppose we are given n proper
continuous self maps σi : X → X for 1 ≤ i ≤ n, i.e., a multivariable dynamical system. In this paper, we develop a theory of conjugacy algebras for such multivariable
dynamical systems. One of the goals is to develop connections between the dynamics of multivariable systems and fundamental concepts in operator algebra theory.
One of the main outcomes of this work is that the classiﬁcation and representation
theory of conjugacy algebras is intimately connected to piecewise conjugacy and
generalized recurrence for multivariable systems.
In the case of a dynamical system with a single map σ, there is one natural prototypical operator algebra associated to it, the semicrossed product of the system.
As we shall see, the case n > 1 oﬀers a far greater diversity of examples. It happens that there are various non-isomorphic algebras that can serve as a prototype
for the conjugacy algebra of the system. The algebras should contain an isometric
copy of C0 (X); plus they need to contain generators si which encode the covariance
relations of the maps σi . In addition, it is necessary to impose norm conditions
to be able to talk about a universal operator algebra for the system. The choice
of these conditions creates two natural choices for the appropriate universal operator algebra for the system: the case in which the generators are either isometric,
producing the semicrossed product, or row isometric, producing the tensor algebra.
The main goal of the paper is to demonstrate that these operator algebras
encode (most of) the dynamical system. The strongest possible information that
might be recovered from an operator algebra of the form we propose would be to
obtain the system up to conjugacy and permutation of the maps, since there is
no intrinsic order on the generators. It turns out that what naturally occurs is a
local conjugacy, in which the permutation may change from one place to another.
This leads us to the notion that we call piecewise conjugacy. We will show that
either of our universal operator algebras contains enough information to recover
the dynamical system up to piecewise conjugacy.
These results oﬀer new insights into the classiﬁcation theory for operator algebras. In [38], Muhly and Solel initiated an ambitious program of classifying all
tensor algebras of C*-correspondences up to isomorphism. They introduced a notion of aperiodicity for C*-correspondences, and were able to classify up to isometric
isomorphism all tensor algebras associated with aperiodic correspondences. Many
important operator algebras, including various natural subalgebras of the Cuntz
algebras, were left out of their remarkable classiﬁcation scheme. A ﬁrst eﬀort to
address the periodic case was the study of isomorphisms between graph algebras
[23, 29, 52]. The classiﬁcation results of this paper for tensor algebras of multidimensional systems includes many examples which are not aperiodic, and also
1
2
K. R DAVIDSON, E. G. KATSOULIS
pushes the envelope beyond isometric isomorphisms. The complexity of the arguments involved in our setting, as well as the need for importing non-trivial results
from other ﬁelds of mathematics, seem to indicate that a comprehensive treatment
of the periodic case for arbitrary tensor algebras of C*-correspondences may not be
feasible at this time.
As a ﬁrst step in understanding our operator algebras for multi-variable dynamical systems, we produce concrete models by way of dilation theory. In recent work,
Dritschel and McCullough [15] show that the maximal representations of an operator algebra A are precisely those which extend (uniquely) to a ∗-representation of
the C*-envelope, C∗env (A). They use this to provide a new proof of the existence of
the C*-envelope independent of Hamana’s theory [20] of injective envelopes. When
such representations are irreducible, they are called boundary representations. This
key notion was introduced by Arveson in his seminal work [2] on dilation theory
for operator algebras (which are generally neither abelian nor self-adjoint). Very
recently, Arveson [4] has shown that there are always suﬃciently many boundary
representations to determine C∗env (A).
The dilation theory is both more straightforward and more satisfying in the case
of the tensor algebra. One can explicitly exhibit a natural and tractable family of
boundary representations which yield a completely isometric representation of the
operator algebra. This provides the ﬁrst view of the C*-envelope.
It also turns out that the tensor algebra is a C*-correspondence in the sense of
Muhly and Solel [36]. This enables us to exploit their work, and work of Katsura
[27] and Katsoulis–Kribs [25], in order to describe the C*-envelope of the tensor
algebra as a Cuntz–Pimsner algebra.
In the semicrossed product situation, one needs to work harder to achieve what
we call a full dilation. These are the maximal dilations in this context. This allows
us to show that generally these algebras are not C*-correspondences. We have no
‘nice’ class of representations which yield a completely isometric representation.
So the explicit form of the C*-envelope remains somewhat obscure in this case.
Nevertheless, the information gained from the dilation theory for the semicrossed
product plays an important role in the sequel. Indeed, in Example 3.24 we use
ﬁniteness for the C∗ -envelope to show that the classiﬁcation scheme for tensor
algebras (see below) is not applicable in the semicrossed product situation.
We then turn to the problem of recovering the dynamics from the operator
algebra. As a ﬁrst step, we establish that algebraic isomorphisms between two
algebras of this type are automatically continuous. Then we apply the techniques
from our analysis [11] of the one-variable case to study the space of characters and
the two-dimensional nest representations. These spaces carry a natural analytic
structure which is critical to the analysis. The fact that we are working in several
variables means that we need to rely on some well-understood but non-trivial facts
about analytic varieties in Cn in order to compare multiplicities of maps in two
isomorphic algebras. The conclusion is that we recover the dynamics up to piecewise
conjugacy.
For the converse, we would like to show that piecewise conjugacy implies isomorphism of the algebras.
For the tensor algebra we show that the converse holds for any type of isomorphism provided that either n ≤ 3 or the covering dimension of X is at most
1. We conjecture that this holds in complete generality. This conjecture is backed
1.1. THE ONE VARIABLE CASE
3
up by the analysis of the n = 3 case, in which we require non-trivial topological
information about the Lie group SU (3). The conjectured converse reduces to a
question about the unitary group U (n). While the topology of SU (n) and U (n)
gets increasing complicated for n ≥ 4, there is reason to hope that there is a positive
answer in full generality.
On the other hand, little is known about the converse for semicrossed products. In Example 3.24 we show that unlike the tensor algebra situation, there are
multisystems on totally a disconnected space which are piecewise conjugate and yet
their semicrossed products are not completely isometrically isomorphic. We do not
know however whether this failure can occur at the algebraic isomorphism level.
In Chapter 4, we consider another connection between the operator algebra
and the dynamical system. We characterize when the operator algebra, either
the tensor algebra or the crossed product, is semisimple strictly in terms of the
dynamics. In the case of a single map, the radical of the semicrossed product has
been studied [33, 44] and ﬁnally was completely characterized by Donsig, Katavolos
and Manoussos [13] using a generalized notion of recurrence. Here we introduce a
notion of recurrence and wandering sets for a dynamical system which is appropriate
for a non-commutative multivariable setting such as ours. The main result of this
chapter is the characterization of semisimplicity in these terms.
Finally in the last chapter of this paper we mention some open problems and
further direction for future research.
1.1. The one variable case
There is a long history of associating operator algebras to dynamical systems,
going back to the work of von Neumann in the 1930’s. In the self-adjoint context,
one is dealing with a (generally amenable) group of homeomorphisms. The abstract
notion of a crossed product of a C*-algebra by an automorphism (or group of
automorphisms) is an important general construction. There is a rich history of
associating C*-invariants with the associated dynamical systems.
The use of a nonself-adjoint operator algebras to encode a dynamical system was
ﬁrst introduced by Arveson [1] and Arveson–Josephson [5] for a one-variable system
(X, σ). In their context, σ was a single homeomorphism with special properties.
A concrete representation was built from an appropriate invariant measure. With
certain additional hypotheses, they were able to show that the operator algebra
provided a complete invariant up to conjugacy.
The abstract version of the semicrossed product of a dynamical system (X, σ)
was introduced by Peters [44]. He does not require the existence of good invariant
measures; nor does he require σ to be a homeomorphism. He does require X to
be compact. With this new algebra, Peters was able to show that the semicrossed
product is a complete invariant for the system up to conjugacy assuming that σ
has no ﬁxed points.
In [21], Hadwin and Hoover considered a rather general class of conjugacy
algebras associated to a single dynamical system. Their proofs work in considerable
generality, but the semicrossed product remains the only natural choice for the
operator algebra of a system. Their methods allowed a further weakening of the
hypotheses. The set {x ∈ X : σ 2 (x) = σ(x) = x} should have no interior, but there
is no condition on ﬁxed points. Then again they were able to recover the dynamics,
up to conjugacy, from the operator algebra.
4
K. R DAVIDSON, E. G. KATSOULIS
In [11], we used additional information available from studying the 2-dimensional nest representations of the semicrossed product to completely eliminate the extraneous hypotheses on (X, σ). We now know that the semicrossed product, even
as an algebra without the norm structure, encodes the system up to conjugacy. We
were also able to replace a compact X with a locally compact space and, as a bonus,
we were able to classify crossed products of the disc algebra by endomorphisms.
1.2. Universal operator algebras
We now discuss the choice of an appropriate covariance algebra for the multivariable dynamical system (X, σ). An operator algebra encoding (X, σ) should
contain C0 (X) as a C*-subalgebra, and there should be n elements si satisfying the
covariance relations
f si = si (f ◦ σi )
for
f ∈ C0 (X) and 1 ≤ i ≤ n.
This relation shows that sik fk sik−1 fk−1 . . . si1 f1 = sw g where we write sw = sik sik−1
. . . sik and g is a certain product of the fj ’s composed with functions built from the
σi ’s. Thus the set of polynomials in s1 , . . . , sn with coeﬃcients in C0 (X) forms an
algebra which we call the covariance algebra A0 (X, σ). The universal algebra should
be the (norm-closed non-selfadjoint) operator algebra obtained by completing the
covariance algebra in an appropriate operator algebra norm.
Observe that in the case of compact X, A0 (X, σ) is unital, and will contain the
elements si as generators. When X is not compact, it is generated by C0 (X) and
elements of the form si f for f ∈ C0 (X).
By an operator algebra, we shall mean an algebra which is completely isometrically isomorphic to a subalgebra of B(H) for some Hilbert space H. By the
Blecher–Ruan–Sinclair Theorem [7], there is an abstract characterization of such
algebras. See [6, 43] for a thorough treatment of these issues. Our algebras are
suﬃciently concrete that we will not need to call upon these abstract results. Nevertheless, it seems more elegant to us to deﬁne universal operator algebras abstractly
rather than in terms of speciﬁc representations.
An operator algebra claiming to be the operator algebra of the system must be
universal in some way. This requires a choice of an appropriate norm condition on
the generators. A few natural choices are:
(1) Contractive: si ≤ 1 for 1 ≤ i ≤ n.
(2) Isometric: s∗i si = I for
1 ≤ i ≤ n.
s1 s2 . . . sn ≤ 1.
(3) Row Contractive:
(4) Row Isometric: s1 s2 . . . sn is an isometry;
i.e. s∗i sj = δij for 1 ≤ i, j ≤ n.
One could add variants such as unitary, co-isometric, column contractive, etc.
In the one variable case, all of these choices are equivalent. Indeed, the Sz.Nagy
isometric dilation of a contraction is compatible with extending the representation
of C0 (X). This leads to the semi-crossed product introduced by Peters [44]. Various
non-selfadjoint algebras associated to a dynamical system (with one map) have been
studied [1, 5, 31, 21, 49, 33, 13].
Once one goes to several variables, these notions are distinct, even in the case
of commutative systems. For example, with three or more commuting variables,
examples of Varopoulos [53] and Parrott [41] show that three commuting contractions need not dilate to three commuting isometries. However a dilation theorem
1.2. UNIVERSAL OPERATOR ALGEBRAS
5
of Drury [16] does show that a strict row contraction of n commuting operators
dilates to (a multiple of) Arveson’s d-shift [3]. While this is not an isometry, it is
the appropriate universal commuting row contraction.
For non-commuting variables, where there is no constraint such as commutativity, one could dilate the n contractions to isometries separately. We shall see
that this can be done while extending the representation of C0 (X) to maintain the
covariance relations. Also for the row contraction situation, there is the dilation
theorem of Frahzo–Bunce–Popescu [17, 8, 47] which allows dilation of any row
contraction to a row isometry. Again we shall show that this can be done while
extending the representation of C0 (X) to preserve the covariance relations.
Definition 1.1. A locally compact Hausdorﬀ space X together with n proper
continuous maps σi of X into itself for 1 ≤ i ≤ n will be denoted by (X, σ). We
shall refer to this as a multivariable dynamical system. It will be called metrizable
if X is metrizable.
We now deﬁne the two universal operator algebras which we will associate to
(X, σ). We justify the nomenclature below.
Definition 1.2. Given a multivariable dynamical system (X, σ), deﬁne the
tensor algebra to be the universal operator algebra A(X, σ) generated by C0 (X)
and generators s1 , . . . , sn satisfying the covariance relations
f si = si (f ◦ σi )
f ∈ C0 (X) and 1 ≤ i ≤ n
and satisfying the row contractive condition s1 s2 . . . sn ≤ 1.
Similarly, we deﬁne the semicrossed product to be the universal operator
algebra C0 (X) ×σ F+
n generated by C0 (X) and generators s1 , . . . , sn satisfying the
covariance relations and satisfying the contractive condition si ≤ 1 for 1 ≤ i ≤ n.
for
We will not belabour the set theoretic issues in deﬁning a universal object like
this, as these issues are familiar. Suﬃce to say that one can ﬁx a single Hilbert space
of suﬃciently large dimension, say ℵ0 |X|, on which we consider representations of
C0 (X) and the covariance relations. Then one puts the abstract operator algebra
structure on A0 (X, σ) obtained by taking the supremum over all (row) contractive
representations. Alternatively, one forms the concrete operator algebra by taking
a direct sum over all such representations on this ﬁxed space.
A case can be made for preferring the row contraction condition, based on
the fact that this algebra is related to other algebras which have been extensively
studied in recent years. If X is a countable discrete set, then the row contractive
condition yields the graph algebra of the underlying directed graph that forgets
which map σi is responsible for a given edge from x to σi (x). In the general case,
this turns out to be a C*-correspondence algebra, or tensor algebra, as deﬁned by
Muhly and Solel [36]. It is for this reason that we call this algebra the tensor algebra
of the dynamical system. As such, it sits inside a related Cuntz–Pimsner C*-algebra
[46], appropriately deﬁned and studied by Katsura [27] building on an important
body of work by Muhly and Solel beginning with [36, 37]. This Cuntz-Pimsner
algebra turns out to be the C*-envelope [2, 20] of the tensor algebra [36, 19, 23].
The C*-envelope of the tensor algebra is therefore always nuclear.
We may consider the dynamical system (X, σ) as an action of the free semigroup
+
.
F+
n The free semigroup Fn consists of all words in the alphabet {1, 2, . . . , n} with
the empty word ∅ as a unit. For each w = ik ik−1 . . . i1 in F+
n , let σw denote the
6
K. R DAVIDSON, E. G. KATSOULIS
map σik ◦ σik−1 ◦ · · · ◦ σi1 . This semigroup of endomorphisms of X induces a family
of endomorphisms of C0 (X) by αw (f ) = f ◦ σw . The map taking w ∈ F+
n to αw is
into
End(C
(X));
i.e.
α
α
=
α
for
v,
w
∈ F+
an antihomomorphism of F+
0
v w
wv
n
n.
This leads us to consider the contractive condition, which is the same as considering contractive covariant representations of the free semigroup. Hence we call the
universal algebra the semi-crossed product C0 (X) ×σ F+
n of the dynamical system.
It also has good properties. However we do not ﬁnd this algebra as tractable as
the tensor algebra. Indeed, several problems that are resolved in the tensor algebra
case remain open for the semicrossed product. In particular, it often occurs (see
Proposition 2.22) that the C*-envelope of the semicrossed product is not nuclear.
In both cases, the (row) contractive condition turns out to be equivalent to
the (row) isometric condition. This is the result of dilation theorems to extend
(row) contractive representations to (row) isometric ones. These are analogues of a
variety of well-known dilation theorems. The tensor algebra case is easier than the
semicrossed product, and in addition, there is a nice class of basic representations
in this case that determine the universal norm. Indeed we exhiibit suﬃciently many
boundary representations to explicitly represent the C*-envelope. In the case of the
crossed product, one needs to introduce the notion of a full isometric dilation; and
these turn out to yield the maximal representations of the C*-envelope.
CHAPTER 2
Dilation Theory
2.1. Dilation for the tensor algebra
We ﬁrst consider a useful family of representations for the tensor algebra analogous to those used by Peters [44] to deﬁne the semi-crossed product of a one
variable system.
By Fock space, we mean the Hilbert space 2 (F+
n ) with orthonormal basis {ξw :
w ∈ F+
}.
This
has
the
standard
left
regular
representation
of the free semigroup
n
deﬁned
by
F+
n
Lv ξw = ξvw for v, w ∈ F+
n.
Consider the following orbit representations of (X, σ). Fix x in X. The orbit
of x is O(x) = {σw (x) : w ∈ F+
n }. To this, we identify a natural representation of
A(X, σ). Deﬁne a ∗-representation πx of C0 (X) on the Fock space Fx = 2 (F+
n ) by
πx (f ) = diag(f (σw (x))), i.e.
f ∈ C0 (X) and w ∈ F+
n.
Send the generators si to Li , and let Lx = L1 . . . Ln . Then (πx , Lx ) is easily
seen to be a covariant representation.
Deﬁne the full Fock representation to be the (generally non-separable) repre⊕
⊕
⊕
sentation (Π, L) where Π = x∈X πx and L = x∈X Lx on FX = x∈X Fx . We
will show that the norm closed algebra generated by Π(C0 (X)) and Π(C0 (X))Li
for 1 ≤ i ≤ n is completely isometric to the tensor algebra A(X, σ). When X is
separable, a direct sum over a countable dense subset of X will yield a completely
isometric copy on a separable space.
Now we turn to the dilation theorem, which is straight-forward given our current knowledge of dilation theory. When the dynamical system is surjective, this is
closely related to [36, Theorem 3.3].
πx (f )ξw = f (σw (x))ξw
for
Theorem 2.1. Let (X, σ) denote a multivariable dynamical system. Let π be
a ∗-representation of C0 (X) on a Hilbert space H, and let A = A1 . . . An be
a row contraction satisfying the covariance relations
π(f )Ai = Ai π(f ◦ σi )
for
1 ≤ i ≤ n.
Then there is a Hilbert
H, a ∗-representation ρ of C0 (X) on K
space K containing
and a row isometry S1 . . . Sn such that
(i) ρ(f )Si = Si ρ(f ◦ σi ) for f ∈ C0 (X) and 1 ≤ i ≤ n.
(ii) H reduces ρ and ρ(f )|H = π(f ) for f ∈ C0 (X).
(iii) H⊥ is invariant for each Si , and PH Si |H = Ai for 1 ≤ i ≤ n.
Proof. The dilation of A to a row isometry S is achieved by the Frahzo–
Bunce–Popescu dilation [17, 8, 47]. Consider the Hilbert space K = H ⊗ 2 (F+
n)
where we identify H with H ⊗ Cξ∅ . Following Bunce, consider A as an operator
7
8
K. R DAVIDSON, E. G. KATSOULIS
in B(H(n) , H). Using the Schaeﬀer form of the isometric
dilation, we can write
D = (IH ⊗ In − A∗ A)1/2 in B(H(n) ) and IH ⊗ L = IH ⊗ L1 . . . IH ⊗ Ln . We
(n)
make the usual observation that (Cξ∅ )⊥ is identiﬁed with 2 (F+
in such a way
n)
(n)
that Li |(Cξ∅ )⊥ Li for 1 ≤ i ≤ n.
Then a (generally non-minimal) dilation is obtained as
A
0
S=
JD IH ⊗ L(n)
where J maps H(n) onto H ⊗ Cn ⊂ K where the ith standard basis vector ei in Cn
is sent to ξi . Then
Ai
0
Si =
(n)
JDi IH ⊗ Li
where Di = D|H ⊗ Cei is considered as an element of B(H, H(n) ).
To extend π, deﬁne a ∗-representation ρ on K by
ρ(f ) = diag(π(f ◦ σw )).
That is,
ρ(f )(x ⊗ ξw ) = π(f ◦ σw )x ⊗ ξw for x ∈ H, w ∈ F+
n.
n
The restriction ρ1 of ρ to H ⊗ C is just ρ1 (f ) = diag(π(f ◦ σi )). The covariance
relations for (π, A) may be expressed as
π(f )A = Aρ1 (f ).
From this it follows that ρ1 (f ) commutes with A∗ A and thus with D. In particular,
ρ1 (f )Di = Di π(f ◦ σi ). The choice of J then ensures that
ρ(f )Si |H⊗Cξ∅ = Si |H⊗Cξ∅ π(f ◦ σi ).
But the deﬁnition of ρ shows that
ρ(f )(IH ⊗ Li ) = (IH ⊗ Li )ρ(f ◦ σi )
Hence, as Si agrees with IH ⊗ Li on H⊥ = H ⊗ (Cξ∅ )⊥ , we obtain
ρ(f )Si |H⊥ = Si π(f ◦ σi )|H⊥ = Si |H⊥ π(f ◦ σi )|H⊥ .
Combining these two identities yields the desired covariance relation for (ρ, S).
The other properties of the dilation are standard.
Remark 2.2. If one wishes to obtain the minimal dilation, one restricts to the
smallest subspace containing H which reduces ρ and each Si . The usual argument
establishes uniqueness.
Corollary 2.3. Every row contractive representation of the covariance algebra
dilates to a row isometric representation.
We now relate this to the orbit representations. It was an observation of Bunce
[8] that the dilation S of A is pure if A = r < 1, where pure means that S is
a multiple of the leftregular representation L. In this case, the range N0 of the
projection P0 = I − ni=1 Si Si∗ is a cyclic subspace for S.
Observe that for any f ∈ C0 (X),
∗
ρ(f )Si Si∗ = Si ρ(f ◦ σi )Si∗ = Si Si ρ(f ◦ σi )
∗
= Si ρ(f )Si = Si Si∗ ρ(f ).
2.2. BOUNDARY REPRESENTATIONS AND THE C*-ENVELOPE
9
So P0 commutes with ρ. Deﬁne a ∗-representation of C0 (X) by ρ0 (f ) = ρ(f )|N0 .
⊕ Then we can recover ρ from ρ0 and the covariance relations. Indeed, K =
Nw where Nw = Sw N0 . We obtain
w∈F+
n
∗
∗
ρ(f )PNw = ρ(f )Sw P0 Sw
= Sw ρ(f ◦ σw )P0 Sw
∗
= Sw ρ0 (f ◦ σw )Sw
.
The spectral theorem shows that ρ0 is, up to multiplicity, a direct integral of point
evaluations. Thus it follows that the representation (ρ, S) is, in a natural sense,
the direct integral of the orbit representations. Thus its norm is dominated by the
norm of the full Fock representation.
As a consequence, we obtain:
Corollary 2.4. The full Fock representation is a faithful completely isometric
representation of the tensor algebra A(X, σ).
sw fw belongs to A0 (X, σ) (i.e. fw = 0
Proof. By deﬁnition, if T = w∈F+
n
except ﬁnitely often), its norm in A(X, σ) is determined as
Aw π(fw )
T σ := sup
w∈F+
n
over the set of all row contractive representations (π, A). Clearly, we can instead
sup over the set (π, rA) for 0 < r < 1; so we may assume that A = r < 1. Then
arguing as above, we see that (π, A) dilates to a row isometric representation (ρ, S)
which is a direct integral of orbit representations. Consequently the norm
Aw π(fw ) ≤
Sw ρ(fw ) ≤
Lw Π(fw ).
w∈F+
n
w∈F+
n
w∈F+
n
Thus the full Fock representation is completely isometric, and in particular is faithful.
Remark 2.5. Indeed, the same argument shows that a faithful completely
isometric representation is obtained whenever ρ0 is a faithful representation of
C0 (X). Conversely a representation ρ0 on H induces a Fock representation ρ on
K = H ⊗ 2 (F+
n ) by ρ(f ) = diag(ρ0 (f ◦ σw )). Then sending each si to IH ⊗ Li yields
a covariant representation which is faithful if ρ0 is.
2.2. Boundary representations and the C*-envelope
As mentioned in the Introduction, we are interested in the maximal dilations.
A completely contractive representation ρ of an operator algebra A on a Hilbert
space H is maximal if, whenever π is a completely contractive dilation of ρ on
a Hilbert space K = H ⊕ K1 , then H reduces π, whence π decomposes as π =
ρ ⊕ π1 . Such representations have the unique extension property: if we consider
A as a subalgebra of a C*-algebra A generated by any completely isometric image
of A, then there is a unique completely positive extension of ρ to A, and it is
a ∗-representation. Such representations always factor through the C*-envelope,
C∗env (A). In particular, if one has a completely isometric maximal representation ρ
of A, then C∗env (A) = C∗ (ρ(A)). The maximal representations which are irreducible
(no reducing subspaces) are called boundary representations.
10
K. R DAVIDSON, E. G. KATSOULIS
It follows from [4] that there are suﬃciently many boundary representations; so
that their direct sum yields a completely isometric representation of A, producing
the C*-envelope. We will exhibit such representations explicitly for A(X, σ).
Lemma 2.6. Suppose that ρ is a completely contractive representation of
A(X, σ) such that each Si = ρ(si ) is an isometry and
n
n
Si Si∗ = Eρ
σi (X) ,
i=1
i=1
where Eρ denotes the spectral measures associated to ρ(C0 (X)). Then ρ is maximal.
Proof. Let ρ be a representation of A(X, σ) on a Hilbert space H; and suppose
that π is any dilation on a space K = H ⊕ K1 . The restriction of π to C0 (X) is
a ∗-representation. As H is invariant, it must reduce π(C0 (X)). So it suﬃces to
show that H also reduces each π(si ).
Since f si = si (f ◦ σi ), it follows that f si = 0 whenever f vanishes on σi (X).
Therefore π(si ) = Eπ (σi (X))π(si ). Since these isometries have orthogonal range,
we always have
n
n
π(si )π(si )∗ ≤ Eπ
σi (X) .
i=1
By hypothesis,
n
i=1
ρ(si )ρ(si )∗ = Eρ
i=1
n
σi (X) ,
i=1
Therefore π(si )K1 will be orthogonal to
n
n
Eρ
σi (X) H + Eπ X \
σi (X) K,
i=1
which contains H. Hence H is reducing.
i=1
By the results of the previous section, it suﬃces to ﬁnd irreducible maximal
dilations of the orbit representations in order to have enough boundary representations to determine the C*-envelope.
To do this, we need to recall the classiﬁcation of atomic representations of the
Cuntz and Cuntz–Toeplitz algebras from [9]. An atomic representation π of En
on a Hilbert space H with a given orthonormal basis {en } is given by n isometries
Si = π(si ) with orthogonal ranges which each permute the basis up to multiplication
by scalars in the unit circle. The irreducible atomic representations of En split into
three types:
(1) The left regular representation λ of F+
n.
(2) The inﬁnite tail representations, which are inductive limits of λ. These are
obtained from an inﬁnite sequence i = i0 i1 i2 . . . in the alphabet {1, . . . , n}. For
s
each s ≥ 0, let Gs denote a copy of Fock space with basis {ξw
: w ∈ F+
n }. Identify Gs
s+1
s
with a subspace of Gs+1 via Ris , where Rj ξw = ξwj . Set πs to be the representation
λ on Gs . Since Rj commutes with λ, we obtain πs+1 Ris = Ris πs . So we may deﬁne
πi to be the inductive limit of the representations πs . It is clear that the sum of
the ranges of πi (si ) is the whole space; so this yields a representation of On . πi is
irreducible if and only i is not eventually periodic; and two are unitarily equivalent
if and only if they are shift–tail equivalent, meaning that after deleting enough
initial terms from each sequence, they then coincide.
2.2. BOUNDARY REPRESENTATIONS AND THE C*-ENVELOPE
11
(3) The ring representations. These are given by a word u = i1 . . . ik and
λ ∈ T. Let Ck be the cyclic group with k elements; and let Ku be a Hilbert space
with orthonormal basis {ξs,w : s ∈ Ck , w ∈ Fn \ Fn is }. Deﬁne a representation τu,λ
of Fn by
τu,λ (si )ξs+1,∅ = λξs,∅
τu,λ (si )ξs,w = ξs,iw
if
i = is
if
|w| ≥ 1 or i = is .
This representation is irreducible if and only if u is primitive (not a power of a
smaller word); and another such representation τv,μ is unitarily equivalent if and
only if v is a cyclic permutation of u and λk = μk .
To state the theorem, we need to deﬁne some analogues that generalize the
orbit representations.
(1) The ﬁrst type are the orbit representations themselves.
(2) An inﬁnite tail representation is given by an inﬁnite sequence i = i0 i1 i2 . . .
in the alphabet {1, . . . , n} and a corresponding sequence of points xs ∈ X for s ≥ 0
such that σis (xs+1 ) = xs . With the setup as in (2) above, we associate each bas
with the point xsw := σw (xs ) in X. Observe that by construction,
sis vector ξw
σwis (xs+1 ) = σw (xs ); so that these points are well deﬁned. Deﬁne a representation
s
s
= f (xsw )ξw
. This is eviπi by deﬁning it on the si as above, and setting πi (f )ξw
dently the inductive limit of the orbit representations πxs ; and thus is a completely
contractive representation of A(X, σ).
(3) A ring representation is given by a word u = i1 . . . ik , a scalar λ ∈ T, and
a set of points xs ∈ X for s ∈ Ck satisfying σis (xs+1 ) = xs . Again we associate
a point in X to each basis vector ξs,w by settng xs,w := σw (xs ). We deﬁne the
representation τu,λ on the si as above; and set τu,λ (f )ξs,w = f (xs,w )ξs,w . It is
routine to verify that this is a representation of A(X, σ).
We can now state the result we want.
Theorem 2.7. The following are all boundary representations of the tensor
algebra A(X, σ).
(1) An orbit representation πx for a point x in X \ ni=1 σi (X).
(2) An inﬁnite tail representation πi given by an inﬁnite sequence i = i0 i1 i2 . . .
in the alphabet {1, . . . , n} and a corresponding sequence of distinct points
xs ∈ X for s ≥ 0 such that σis (xs+1 ) = xs .
(3) A ring representation τu,λ given by a word u = i1 . . . ik , a scalar λ ∈ T
and a set of distinct points xs ∈ X for s ∈ Ck satisfying σis (xs ) = xs+1 .
Proof. First let us verify that thatthese representations are maximal. This
is immediate from Lemma 2.6 because ni=1 ρ(si )ρ(si )∗ is the identity in the last
∗
two cases, and is I − ξ∅ ξ∅
in the ﬁrst case. This is the only non-trivial case,
∗
≤
but here the hypothesis
that
x is not in the range of any σi means that ξ∅ ξ∅
n
Eπx X \ i=1 σi (X) . Indeed this is an equality, as by construction, every other
basis vector corresponds to a point in the orbit of x; and thus lies in the range of
Eπx (σi (X)) for some i.
It remains to verify that these representations are irreducible. The ﬁrst type is
irreducible because the restriction to the algebra generated by s1 , . . . , sn is the left
regular representation, and this restriction is already irreducible.
In case (3), it follows from [9] that the projection P onto the ring space
span{ξs,∅ : s ∈ Ck } lies in the wot-closed algebra generated by s1 , . . . , sn ; and

Author Kenneth R. Davidson Isbn 9780821853023 File size 953.95KB Year 2011 Pages 53 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $sigma_i:X to X$ for $1 le i le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $mathcal{A}(X,tau)$ and the semicrossed product $mathrm{C}_0(X)times_taumathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.|Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $sigma_i:X to X$ for $1 le i le n$. To this the authors associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $mathcal{A}(X,tau)$ and the semicrossed product $mathrm{C}_0(X)times_taumathbb{F}_n^+$. They develop the necessary dilation theory for both models. In particular, they exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra. Download (953.95KB) Index Theory for Locally Compact Noncommutative Geometries Infinitesimal Geometry of Quasiconformal and Bi-lipschitz Mappings in the Plane Decorated Teichmuller Theory Geometric Function Theory and Non-linear Analysis Operator-Valued Measures, Dilations, and the Theory of Frames Load more posts