Download 2.2 The abstract Toeplitz algebra

Chapter 2. Examples of Banach and C -algebras
78
2.2
The abstract Toeplitz algebra
2.2.1 Enveloping C-algebra
Let A be a -algebra, that is, an algebra equipped with an anti-linear, antimultiplicative involution a ÞÑ a . In general, the resulting notion of positivity
for elements in A °
can be pathological, for example by the existence of relations of the form nk1 ak ak 0. We will however be mostly interested in
-algebras which admit a faithful representation on a Hilbert space.
1. Let CrX s be the polynomial algebra in one variable,
equipped with the unique -algebra structure such that X X . Let H
be a separable Hilbert space, and x P B pHq a self-adjoint element with
innite spectrum. Then we have an injective unital -homomorphism
Examples 2.2.1.
πx : CrX s Ñ B pHq,
ppX q ÞÑ ppxq.
The closure of πx pCrX sq is the C -algebra generated by idH and x, and
hence isomorphic to Cb pSppxqq.
2. Let Γ be a discrete group, and CrΓs the group algebra of Γ. Denote the
natural basis elements of CrΓs by λg . Then CrΓs is a -algebra with
the -structure uniquely determined by λg λg1 . We then have an
injective unital -homomorphism, the regular representation,
πreg : CrΓs Ñ B pl2 pΓqq,
λg
ÞÑ ug ,
where pug f qphq f pg 1 hq for f P l2 pΓq. The closure of πreg pCrΓsq, that
is, the C -algebra generated by the ug , is called the reduced C -algebra
pΓq of Γ.
Cred
Another question one can pose for -algebras is the following: given a algebra A, when does there exist a C -algebra B and a -homomorphism
πu : A Ñ B which is a solution to the universal problem of -representing A
in C -algebras? That is, pB, πu q should be such that for any C -algebra C
and any -homomorphism π : A Ñ C , there exists a unique -homomorphism
Π : B Ñ C such that Π πu π .
2.2 The abstract Toeplitz algebra
B
πu
79
/
π
?C
(2.2)
Π
A
In this case, we say A admits a universal C -envelope, namely, the C -algebra
B . It is then unique up to -isomorphism by abstract nonsense. Note that
we do not require the associated homomorphism πu : A Ñ B to be injective.
Lemma 2.2.2. Let A be a -algebra with generating set S (so any element
in A is a polynomials expression in the elements of S Y S ). Then A admits
a C -envelope if and only if for every x P S , there exists Cx ¥ 0 such that
for any -representation π : A Ñ B pHq, one has }πpxq} ¤ Cx .
Proof. Assume rst that A admits a universal C -envelope pB, πu q. Then
any -homomorphism π : A Ñ B pHq extends to a -representation Π : B Ñ
B pHq. As a -representation of a C -algebra is contractive, it follows that
}πpxq} ¤ }πupxq} for any x P A.
Conversely, assume for each x P S the existence of Cx ¥ 0 such that }π pxq} ¤
Cx for all -representations π . Then clearly such Cx 's also exist for linear
combinations of products in the elements of S and S . As S is generating,
we deduce that such Cx 's exist for all x P A.
We may assume that the Cx are minimal with respect to this condition. Then
in fact for each x we can nd a -representation πx on a Hilbert space Hx with
}πxpxq} Cx. Indeed, we can take πx `nπn where πn is a -representation
with }πn pxq} ¥ Cx n1 .
Dene then πu `xPA πx on the Hilbert space H `xPA Hx . Let B be the
closure of πu pAq. Then }πu pxq} Cx .
We claim that pB, πu q is the universal C -envelope of A. Indeed, if C is a
C -algebra, we may assume that C „ B pG q for some Hilbert space G . If
then π : A Ñ C is a -homomorphism, it follows that }π pxq} ¤ Cx }πu pxq}
for every x P A. This implies that there exists a unique -homomorphism
Π : B Ñ C such that Πpπu pxqq π pxq for all x P A.
Examples 2.2.3. 1. Let A CrX s with X X . Then A does not
admit a universal C -algebra envelope. Indeed, for each self-adjoint
Chapter 2. Examples of Banach and C -algebras
80
P B pHq, there exists a -representation πx of A on H with
πx pX q x. Hence πx ÞÑ }πx pX q} is unbounded.
element x
2. Let A be a Banach -algebra. By Proposition 1.1.38, it follows that A
admits a universal C -envelope.
3. Let Γ be a discrete group. Then l1 pΓq is a Banach -algebra, cf. Example 1.1.36.5. It follows that l1 pΓq admits a universal C -envelope
Cu pΓq, called also the universal group C -algebra of Γ. Note that we
have a factorisation
pΓq
πred : Cu pΓq Ñ Cred
to the reduced C -algebra from Example 2.2.1.2. In general however,
this map is not injective. We will come back to this in a later chapter.
Remark 2.2.4. One can also form the universal C -algebra of a -algebra
with given generators and a priori norm estimates. Namely, in Lemma 2.2.2
one then only considers those representations in which the norm estimate is
satised. For example, the universal C -envelope of CrX s with the normcondition }X } ¤ 1 is the C -algebra Cb pr1, 1sq. Similarly, one can add the
assumption of a priori positivity for elements.
2.2.2 The abstract Toeplitz C-algebra
Denition 2.2.5. The abstract Toeplitz -algebra T is the unital -algebra
generated7 by the single element s and the single relation s s 1.
Lemma 2.2.6. The abstract Toeplitz -algebra T admits a universal C envelope.
Proof. Let π : T Ñ B pHq be a -representation. Then p πp1q is a selfadjoint projection, and moreover π psq π psq p. It follows from Lemma 2.2.2
that T admits a universal C -envelope.
Denition 2.2.7. The abstract Toeplitz C -algebra T is dened as the universal C -envelope of T .
7 We recall that a unital
-algebra A
B „ A if any element of A
t1u Y B Y B .
is generated by a set
can be written as a polynomial expression in elements of
2.2 The abstract Toeplitz algebra
81
Clearly, there exists a unital -homorphism
T Ñ C pS q „ B pl2 pNqq,
s ÞÑ S
into the concrete Toeplitz algebra. It then follows that it extends to a surjective -homomorphism
T Ñ C pS q.
Our aim is to prove that this is in fact a -isomorphism. This is equivalent to
showing that C pS q is a universal solution to the C -homomorphism problem
for T . This, in turn, is implied by the following.
Theorem 2.2.8 (Coburn's theorem). Let H be a Hilbert space, and U P H
an isometry. Then there exists a unique unital -representation
π : C pS q Ñ B pH q,
S
ÞÑ U.
Moreover, if U U idH , then π is injective.
Proof. Of course unicity is clear, so it suces to show existence.
Let X be a set, and consider the Hilbert space HX
the isometry
"
SX : HX
Ñ HX ,
f
l2pX Nq. Let SX be
x, 0q 0
ÞÑ pSX f q : ppSSXX ff qpqpx,
nq f px, n 1q,
n ¥ 1.
Then under the natural identication
HX
`xPX l2pNq,
the operator SX corresponds to `xPX S . Hence we have a unital -isomorphism
C pS q C pSX q sending S to SX .
On the other hand, let H be a Hilbert space and u a unitary. Then by
the spectral theorem, the inclusion Sppuq „ S1 gives a -homomorphism
Cb pS1 q Ñ C puq sending the identity function to u. Composed with the
natural map C pS q Ñ Cb pS1 q, we obtain a -representation C pS q Ñ B pHq
sending S to u.
Let now U be a general isometry. To prove the theorem, it suces to nd a
set X , a Hilbert space G , a unitary u P B pG q and a unitary
v : H Ñ l 2 pX
Nq ` G
82
such that
Chapter 2. Examples of Banach and C -algebras
vU v SX ` u.
Such a decomposition is known as a Wold decomposition.
Let X be a set of cardinality dimpKerpU qq (which by the above we may
assume ¡ 0, else U is a unitary and we're done). Fix an orthonormal basis
tex | x P X u of KerpU q. If n ¡ m and ξ, η P KerpU q, then U nξ K U mη,
since
xU nξ, U mηy xξ, pU qnU mηy
xξ, pU qnmηy
Hence we can dene an isometry
v 1 : l2 pX Nq Ñ H,
f
0.
ÞÑ
¸
P
P
f px, nqU n ex .
x X,n N
Let G be the orthogonal complement of the range of v 1 . Then
v : l2 pX Nq ` G Ñ H, ξ ` η ÞÑ v 1 ξ η
clearly is a unitary.
We claim that G is invariant under U and U , and that moreover the restriction u of U to G is a unitary. Indeed, we have η P G if and only if η K U n ξ
for any n ¥ 0 and any ξ P KerpU q. Hence G is invariant under U and U .
By construction, KerpU q X G 0. It follows that also u is an isometry,
hence u a unitary.
It is now clear that v U v SX ` u.
2.2.3 Quantum discs and elds of C-algebras
The Toeplitz C -algebra T is generated by a single element S with SppS q D, the closed unit disc. One might say that T is a non-commutative algebra
of functions on the disc, or even more provocatively, an algebra of functions
on a quantum disc. In the following, we are going to give some more meaning
to this statement. First of all, let us show that one can interpolate between
the `quantum disc' and the real disc.
2.2 The abstract Toeplitz algebra
83
Denition 2.2.9. Let 0 ¤ q 1. The quantum disc algebra at parameter
q is the unital -algebra Tq generated by the single element sq with the single
relation
sq sq qsq sq
1 q.
We see that T0 T .
Lemma 2.2.10. The quantum disc algebra Tq admits a universal C -envelope
Tq , and then }sq } 1.
Proof. Let π be a (unital) -representation of Tq , and put y πpsq q. Then
y y p1 q q qyy . It follows that (with rpq the spectral radius)
}y }2 } y y }
rpyyq
rpp1 qq qyyq
p1 qq qrpyyq
p1 qq q}yy}
p 1 q q q }y }2 .
Hence p1 q q}y }2 p1 q q, and }y } 1. As Tq
lemma follows.
is generated by sq , the
The denition of Tq would of course also make sense for q 1, which gives
the universal (commutative) -algebra generated by a single normal element
s1 . However, this clearly does not admit a universal C -algebra. But if we
consider the universal problem with respect to -representations sending s1
to a contractive operator, then clearly the universal C -algebra solution to
this problem is precisely the C -algebra Cb pDq.
We aim to show now that in fact Tq
for the underlying Tq .
T
for any 0 ¤ q 1. This is not true
Let us rst nd a concrete realization of Tq .
Denition 2.2.11. Let 0 ¤ q 1, and let ten | n P Nu denote the standard
orthonormal basis of l2 pNq. Dene a weighted shift on l2 pNq by
a
2
2
Sq : l pNq Ñ l pNq,
en
ÞÑ
1 q n 1 en 1 .
We dene the concrete quantum disc C -algebra at parameter q to be C pSq q.
84
Chapter 2. Examples of Banach and C -algebras
It is clear that Sq e0 0 and
Sq en
a
1 q n en1 ,
n ¡ 0,
and that Sq satises the quantum disc algebra relation
Sq Sq qSq Sq
p1 qqidl pNq.
Proposition 2.2.12. One has C pSq q C pS q, and πQ pSq q πQ pS q in the
2
Calkin algebra.
Proof. Let Tq 1 Sq Sq . Then Tq en qn 1 en , hence Tq P B0 pl2 pNqq. Since
Sq S p1 Tq q1{2 , it follows that πQ pS q πQ pSq q.
Now B0 pl2 pNqq „ C pS q, hence Sq P C pS q and so C pSq q „ C pS q. Conversely, since Tq P C pSq q and p1 Tq q1{2 invertible, we deduce S P C pSq q.
We conclude C pS q C pSq q.
Theorem 2.2.13. All Tq with 0 ¤ q 1 are -isomorphic C -algebras.
Proof. Let us show that the canonical map πq : Tq Ñ C pSq q sending sq to
Sq is a -isomorphism. The theorem will then follow by Proposition 2.2.12.
By Coburn's theorem, we may suppose that q ¡ 0. By the non-commutative
Gelfand-Neumark theorem, we may also assume that Tq „ B pHq for some
Hilbert space H. Write tq 1 sq sq . We then claim that Spptq q „ r0, q s.
Indeed, let us rst show that Sppsq sq q „ t0, 1 q, 1 q 2 , . . .u Y t1u. Suppose
this were not the case. Then we can nd λ P Sppsq sq q with λ 1 and
λ 1 q n for all n P N. But since Sppxy q Y t0u Sppyxq Y t0u for any
couple of operators x, y , we deduce from the dening relation for sq that also
q 1 pλ p1 q qq is a non-zero element in Sppsq sq q. By induction, we obtain
that q n pλ 1q 1 P Sppsq sq q for all n P N. But q n pλ 1q 1 0 for n
large enough, giving a contradiction with Sppsq sq q „ r0, 8q.
Hence also Sppsq sq q „ t0, 1q, 1q 2 , . . .uYt1u. Now if 0 P Sppsq sq q, it follows
from the denining relation of sq that p1 q 1 q P Sppsq sq q, contradicting the
positivity of the operator sq sq . Hence Sppsq sq q „ t1 q, 1 q 2 , . . .u Y t1u.
We deduce that indeed Spptq q „ r0, q s, hence 1 tq is invertible inside Tq .
Dene then s sq p1 tq q1{2 . By its denition, s is an isometry. By Coburn's
2.2 The abstract Toeplitz algebra
85
theorem, we nd that we have a -homomorphism
C pS q Ñ Tq ,
S
ÞÑ s.
As clearly πq psq S (under the isomorphism of Proposition 2.2.12), it follows
that πq is bijective.
Although the Tq are all C -isomorphic, one would like to say Tq is more
commutative than Tq1 if q ¡ q 1 . To substantiate this, one would like to have
a more natural way of comparing dierent Tq when the parameter q varies.
The notion of eld of C -algebras is ideally suited for this.
Denition 2.2.14. Let X be a compact Hausdor space. A Cb pX q-algebra
is a unital C -algebra A together with a unital inclusion
π : Cb pX q „ Z pAq,
where Z pAq is the center of A.8
For x P X , the ber C -algebra Ax at x is dened as the quotient Ax A{Jx
where Jx is the closed 2-sided ideal in A generated by the functions in Cb pX q
which vanish at x P X . We call the canonical map πx : A Ñ Ax the evaluation
map at x.
One can thus interpret a Cb pX q-algebra A as consisting of an X -parametrized
collection of C -algebras Ax . A map x ÞÑ ax P Ax is then considered a
`continuously varying' family of elements in the Ax if and only if there exists
a P A with πx paq ax for all x P X . We say that the assignment x ÞÑ Ax
forms a eld of C -algebras.
Lemma 2.2.15. Let X be a compact space, and A a Cb pX q-algebra. Then
for all a P A, the map
x ÞÑ }πx paq}
is upper semi-continuous.
P A. We are
lim supα }πx paq} ¤ }πx paq}.
Proof. Let a
to prove that for any net xα
Ñ
x, one has
α
8 We recall that the
x P A.
center
of an algebra
A
is the set of
z
P A with zx xz for all
Chapter 2. Examples of Banach and C -algebras
86
Fix x P X . Take ε ¡ 0. By denition of the quotient norm, we can nd
nitely many f°
i P Cb pX q and ai P A such that fi pxq 0 for all i and such
that, with t i fi ai , one has
}πxpaq} ¥ }a t} ε.
Now for each δ ¡ 0, we can nd an open neigborhood U of x and a function
g P Cb pX q such that
a) }g } ¤ 1,
b) g py q 1 for y P U ,
c) }gfi } δ for all i.
°
It then follows that }gt} ¤ δ i }ai }. Hence by choosing δ small enough, we
can assume }gt} ¤ ε. But then
}πxpaq} ¥ }a t} ε
¥aq }gpa tq} ε
¥ }ga} 2ε
}gt}¤ε
}a p1 gqa} 2ε.
But since g py q 1 for all y P U , we have p1 g qpy q 0. So we conclude by
denition of πy that
}πxpaq} ¥ }πy paq} 2ε,
@y P U.
This proves the upper semi-continuity.
In general however, there is no reason why the map x ÞÑ }πx paq} should be
continuous.
Example 2.2.16. Consider X r1, 1s and Y
A Cb pX
r1, 8q. Let
Y q.
Then A is a Cb pX q-algebra by the map Cb pX q Ñ Cb pX Y q associated to
the projection X Y Ñ X . Let φ be a continuous bounded positive function
such that φpx, y q 0 for x 0 and φpx, y q 1 for x ¡ 0 and y ¡ x1 (such
2.2 The abstract Toeplitz algebra
87
a function clearly exists). Then in the ber Ax with x ¡ 0, we have that
πx pφq is a positive element with 1 inside its spectrum, hence }πx pφq} ¥ 1. By
upper semicontinuity, we conclude also }π0 pφq} ¥ 1. But clearly }πx pφq} 0
for x 0. Hence x ÞÑ }πx pφq} is not lower semicontinuous in 0.
Hence, to avoid such pathological situations, we often impose that the above
normfunction should be continuous.
Denition 2.2.17. Let X be a compact Hausdor space, and A a Cb pX qalgebra. We say that the assignment x ÞÑ Ax forms a continuous eld of
C -algebras if and only if for each a P A, the map
x ÞÑ }πx paq}
is continuous.
A general way to produce continuous Cb pX q-algebras is the following.
Proposition 2.2.18. Let X be a compact Hausdor space. Let A be a
Cb pX q-algebra, and suppose there exist states ϕx on Ax such that
a) For all x, the GNS-representation πϕx of Ax is injective.
b) For all a P A, the function x ÞÑ ϕx pax q is continuous.
Then x ÞÑ Ax is a continuous eld of C -algebras.
Proof. As the πϕx are injective, it follows that they are isometric. As the
GNS-vectors ξϕx are cyclic for Ax (and as the maps A Ñ Ax are surjective),
we have for all a P A and x P X that
}ax} supt|xbx ξϕx , πϕx pax qcx ξϕx y | b, c P A, }bx ξϕx } ¤ 1, }cx ξϕx } ¤ 1u
supt|ϕx pbx ax cx q | b, c P A, ϕx pbx bx q ¤ 1, ϕx pcx cx q ¤ 1u.
Pick now x P X , a
1, ϕx pc cx q ¤ 1 and
P A and 0 ε 1.
Choose b, c
P A with ϕxpbxbxq ¤
}ax} ¤ |ϕxpbxaxcxq| ε.
By continuity of x ÞÑ ϕx pfx q for all f P A, we can nd an open neighborhood
U of x such that for all y P U , ϕy pby by q ¤ 1 ε, ϕy pcy cy q ¤ 1 ε and
}ax} ¤ |ϕy pby ay cy q| 2ε.
x
88
Chapter 2. Examples of Banach and C -algebras
Hence
It follows that x
Lemma 2.2.15.
}ax} ¤ }ay }p1 εq 2ε
¤ }ay } εp}a} 2q.
Þ }ax} is lower semi-continuous,
Ñ
hence continuous by
Let us now show that the Toepliz algebras Tq can be collected into a eld of
C -algebras for 0 ¤ q ¤ 1, where we dene T1 Cb pDq (with D the closed
unit disc). In T1 , we dene s1 as the identity function z ÞÑ z , and we dene
T1 as the -algebra generated by s1 .
Proposition 2.2.19. Let I r0, 1s. Let A be the unital -algebra generated
by elements Q, T with Q Q, QT T Q and
T T
QT T 1 Q.
Then A admits a universal C -envelope A with respect to the extra conditions
}T } ¤ 1 and 0 ¤ Q ¤ 1.
Moreover, the assigment p ÞÑ ppQq on polynomials admits an extension to an
imbedding Cb pI q Ñ A, and the associated Cb pI q-algebra A satises Aq Tq .
Proof. Since we impose norm restrictions on generators of A , it is clear that
the universal C -envelope A as above exists. Moreover, from the univeral
relations of A , we conclude that there exists a unital -homomorphism
π q : A Ñ Tq ,
Q ÞÑ q,
T
ÞÑ sq .
As 0 ¤ Q ¤ 1 in A, we also have a -homomorphism
Cb pI q Ñ A,
f
Since πq pf q f pq q, we conclude that f
Cb pI q-algebra.
Þ f pQq.
Ñ
ÞÑ f pQq is injective.
Hence A is a
Let now Iq Kerpπq q. Then clearly f pQq P Iq if f pq q 0. It follows that
Jq „ Iq , with Jq „ A the ideal generated by functions in Cb pX q vanishing at
q . On the other hand, since Q q P Jq , it follows that A{Jq is generated by
2.2 The abstract Toeplitz algebra
a single element Tq
a map
T
89
Jq satisfying Tq Tq qTq Tq
Tq
Ñ A{Jq ,
sq
1 q. Hence we have
ÞÑ Tq .
As this is clearly an inverse to the projection map πq : Aq
that Iq Jq .
Ñ Tq , we conclude
We want to show that the eld q ÞÑ Tq obtained from the Cb pI q-algebra A is
continuous. We will need some preparations.
Denition 2.2.20. For 0 ¤ q 1, dene ϕq as the state
ϕq : Tq
Ñ C,
x ÞÑ p1 q q
8̧
q n xen , xen y.
n 0
For q 1, let ϕ1 be the state associated to the normalized Lebesgue measure
on D.
Remark that ϕq is obviously well-dened since q n is absolutely summable. It
is then easily seen to be a state.
Lemma 2.2.21. Let 0 ¤ q ¤ 1 and x P Tq . Then
ϕq psq xq qϕq pxsq q.
Proof. For q 1 this is obvious. For 0 ¤ q 1, we compute
ϕ q p s q xq
p1 qq
8̧
8̧
q n xen , sq xen y
n 0
p1 q q
8̧
q n xsq en , xen y
n 0
p1 qq p1 qnq1{2qnxen1, xeny
8̧
n 1
p1 qq p1 qn 1q1{2qn 1xen, xen 1y
8̧
n 0
qp1 qq qnxen, xsq eny
n0
qϕq pxsq q.
Chapter 2. Examples of Banach and C -algebras
90
Lemma 2.2.22. For 0 ¤ q 1 and m, n P N, we have
1q
.
1 qn 1
n
n
ϕq psm
q psq q q δm,n q
We follow here the `convention' that 00 1.
Proof. First remark that from the dening relation sq sq qsq sq
follows by induction that
psq qnsq p1 qnqpsq qn1
q n sq psq qn .
1 q, it
(2.3)
n
Now from the denition of ϕq , it is clear that ϕq psm
q psq q q 0 when m n.
Write then an ϕq psnq psq qn q. From Lemma 2.2.21 and Equation (2.3), we
deduce that
an qϕq psn1 ps qn sq q
q
q
qp1 q qan1 qn 1an
Hence an q 11q q an1 . Since a0 1, we deduce an q n 11qq .
Corollary 2.2.23. For a P A, the map q ÞÑ ϕq paq q is continuous on r0, 1s.
n
n
n 1
n 1
Proof. By norm-density of A in A, and since }ϕq πq } ¤ 1, it suces to
prove the statement for a P A . Now from the commutation relations in A ,
it follows easily that each element in A is a linear combination of products
Qk T l pT qm for k, l, m P N. It follows that it is sucient to show that the
functions
q
are continuous for l, m P N.
ÞÑ ϕq pslq psq qmq
Now from Lemma 2.2.22, we see that ϕq pslq psq qm q δm,l q m 11qmq 1 for 0 ¤
q 1. Clearly, this is a continuous function on r0, 1q. It remains to check
continuity in 1. But
»
»
1
1
l
m
l m
ϕ1 ps ps q q z z̄ dz δm,n
|z|2mdz 1 .
1
1
Since limqÑ1 q m 11qmq
π
1
D
π
D
m1 1 , the corollary follows.
m
1
2.2 The abstract Toeplitz algebra
91
Theorem 2.2.24. The eld of C -algebras q ÞÑ Tq induced from the Cb pI qalgebra A is continuous.
Proof. By combining Corollary 2.2.23 with Proposition 2.2.18, it is sucient
to see that the GNS-representations πϕq are injective on Tq for all 0 ¤ q ¤ 1.
Now for 0 q 1 and x P Tq , we have
xξϕ , πϕ pxxqξϕ y p1 qq
q
q
q
¸
P
q n }xen }2 ,
n N
hence πϕq pxq 0 implies x 0.
A similar reasoning works for q 1.
Finally, for q 0, we have for all m, n ¥ 0, that
xsmqξϕ , πϕ pxqsnqξϕ y ϕ0pps0 qmxsn0 q
xsm0 e0, xsn0 e0y
xem, xeny.
0
0
0
We conclude that also πϕ0 is injective.
2.2.4 More on representations of C-algebras
Let A be a unital C -algebra. We have seen that A admits an injective representation A Ñ B pHq on a Hilbert space H. Let us consider the general
representation theory of a C -algebra in more detail.
Let us rst present some general terminology.
Denition 2.2.25. Let A be a unital C -algebra, H1 and H2 two Hilbert
spaces, and π1 and π2 unital -representations on respectively H1 and H2 .
An intertwiner between π1 and π2 is an operator x P B pH1 , H2 q such that
π2 paqx xπ1 paq for all a P A. We denote the set of all intertwiners by
Morpπ1 , π2 q tx P B pH1 , H2 q | x intertwiner between π1 and π2 u.
We say π1 is a subrepresentation of π2 if there exists an isometric intertwiner
between π1 and π2 .
Chapter 2. Examples of Banach and C -algebras
92
We say π1 and π2 are unitarily equivalent if there exists a unitary intertwiner
between π1 and π2 .
Note that if x is an intertwiner from π1 to π2 , then x is an intertwiner from
π2 to π1 . It is also clear that if x P Morpπ1 , π2 q and y P Morpπ2 , π3 q, then
yx P Morpπ1 , π3 q.
Let us say that a closed subspace H1 „ H is invariant under a representation
π of a C -algebra A if π pAqH1 „ H1 .
Denition 2.2.26. Let A be a unital C -algebra. A unital -representation
π of A on a Hilbert space H is called irreducible if H is non-zero and the
only invariant Hilbert spaces in H are t0u and H itself.
1. As π is a -representation, one easily sees that H1 is
invariant if and only if its orthogonal complement is invariant. It follows
that a -representation is irreducible if and only if it is atomic, i.e. can
not be written as a direct sum of two (non-trivial) -representations.
Remarks 2.2.27.
2. If π is irreducible, any non-zero vector ξ P H is necessarily cyclic.
3. As we shall see later, every irreducible representation is even algebraically irreducible: if π is irreducible and t0u „ V „ H is an invariant
linear subspace (not necessarily closed), then either V t0u or V H.
Lemma 2.2.28 (Schur's lemma for C -algebra representations). Let A be a
unital C -algebra.
1. A unital -representation π of A on a Hilbert space H is irreducible if
and only if Morpπ, πq CidH .
2. If π1 and π2 are irreducible unital -representations of A on respective Hilbert spaces H1 and H2 , then either Morpπ1 , π2 q t0u or else
Morpπ1 , π2 q Cu with u a unitary intertwiner.
Proof. Assume π is irreducible, and let x be an intertwiner from π to itself.
We want to show that x is a scalar multiple of the identity operator.
Let us suppose rst that we could nd a normal intertwiner x which is not
a scalar multiple of the identity operator. Then, as x is also an intertwiner
from π to itself, it follows by continuity that each element in C pidH , xq is an
intertwiner of π . Now as x is not a scalar, we can nd, by functional calculus,
two real nonzero functions f, g P Cb pSppxqq such that f g 0. As f pxq 0,
2.2 The abstract Toeplitz algebra
93
the Hilbert space closure of f pxqH is a non-zero Hilbert space H1 . Similarly,
the Hilbert space closure of g pxqH is a non-zero Hilbert space H2 . But since
xf pxqξ, gpxqηy xξ, f pxqgpxqηy
xξ, pf gqpxqηy
0,
H1 is orthogonal to H2 .
It follows that t0u Š H1 Š H. But since f pxq P C pidH , xq is an intertwiner,
we have π paqf pxqξ f pxqπ paqξ for all a P A and ξ P H. We deduce by
continuity that H1 is invariant. This is in contradiction with the assumption
that π is irreducible.
Hence any normal intertwiner is scalar. Let us take now an arbitrary intertwiner x. Then x , x x and xx are also intertwiners. From the rst part
of the proof, it follows that x x and xx are scalars. As x x and xx are
positive operators, these scalars are positive. As x x and xx have the same
norm, we see that in fact these scalars are equal. It follows that x is normal
(and in fact a positive multiple of a unitary). Then by the rst part of the
proof we conclude that x must necessarily be a scalar multiple of the identity.
This proves the rst part of the proposition. To prove the second part,
suppose x is a non-zero intertwiner between π1 and π2 . By the rst part of
the lemma, it follows that x x xx λ for some λ ¡ 0. Hence u λ1{2 x
is a unitary intertwiner between π1 and π2 .
If then y is another intertwiner from π1 to π2 , the rst part of the lemma
gives that u y c for some c P C. Hence y cu.
Assume now that ϕ is a state on a unital C -algebra A. When is the associated GNS-representation πϕ irreducible? It turns out that this property is
determined by the geometrical position of ϕ inside the state space.
Denition 2.2.29. Let A be a unital C -algebra, and S pAq its state space.
We call ϕ P S pAq a pure state if ϕ is an extreme point of S pAq.
Indeed, S pAq is clearly a compact convex set inside the linear space of continuous functionals on A (with the weak- topology). Then, by denition, ϕ is
an extreme point of S pAq if and only if it can not be written as tϕ1 p1 tqϕ2
for two states ϕ1 , ϕ2 with 0 t 1 and ϕ1 ϕ2 .
Chapter 2. Examples of Banach and C -algebras
94
Theorem 2.2.30. Let ϕ be a state on a unital C -algebra A. Then the
GNS-representation πϕ is irreducible if and only if ϕ is extremal.
Proof. Assume that πϕ is not irreducible. Then H H1 ` H2 with H1 and
H2 non-trivial orthogonal invariant subspaces. Let ξϕ ξϕ1
ξϕ2 be the
orthogonal decomposition of ξϕ with respect to this direct sum. Then as ξϕ
is cyclic, it follows that neither ξϕ1 or ξϕ2 are zero. Writing t }ξϕ1 }2 , we have
}ξϕ2 }2 1 t since ξϕ is a unit vector. In particular, 0 t 1. Put
ϕ1 paq 1 1
x
ξϕ , πϕ paqξϕ1 y,
t
ϕ2 paq x
ξϕ2 , πϕ paqξϕ2 y.
1t
1
Then ϕ1 and ϕ2 are states, and ϕ tϕ1 p1 tqϕ2 . If we can show that
ϕ1 ϕ2 , then we can conclude that ϕ is not pure. But if ϕ1 ϕ2 , then
necessarily ϕ1 ϕ. This would mean that for all a P A,
xξϕ, πϕpaqξϕy 1 1
xξ , πϕpaqξϕ1 y
t ϕ
1 1
xξ , πϕpaqξϕy.
t ϕ
By cyclicity of ξϕ , we deduce that ξϕ 1t ξϕ1 , which gives a contradiction.
Conversely, suppose that πϕ is irreducible. Suppose that ϕ were not pure,
say ϕ tϕ1 p1 tqϕ2 for 0 t 1 and states ϕ1 , ϕ2 with ϕ1 ϕ2 . Then
1
1
ϕ1 pa aq tϕ1 pa aq ¤ ptϕ1 pa aq
t
t
p1 tqϕ2paaqq ¤ 1t ϕpaaq.
It follows that we can dene a surjective bounded operator x P B pHϕ , Hϕ1 q
such that
x paξϕ q aξϕ1 ,
@a P A.
As ξϕ is cyclic, it follows from this formula that x is an intertwiner. But as
x x is an intertwiner for πϕ , the irreducibility of πϕ gives that x x is a scalar
λ ¡ 0. Hence for all a P A,
ϕ1 paq xξϕ1 , aξϕ1 y
xxξϕ, x paξϕqy
xξϕ, xx paξϕqy
λxξϕ, aξϕy
λϕpaq.
2.2 The abstract Toeplitz algebra
As ϕ and ϕ1 are both states, this forces λ 1 and hence ϕ
symmetry, also ϕ ϕ2 , contradicting ϕ1 ϕ2 .
95
ϕ1 . By
Let us show now that any C -algebra has enough irreducible representations,
giving a strengthening of the non-commutative Gelfand-Neumark theorem.
Theorem 2.2.31 (Gelfand-Raikov theorem9 ). Let A be a unital C -algebra
and a P A. Then there exists an irreducible unital -representation π of A
with πpaq 0.
Proof. Let χ be the character f pa aq ÞÑ f p}a a}q on B C p1A , a aq. As
χ is a one-dimensional representation, it is clearly irreducible. Hence it is a
pure state on B . Now by Corollary 1.2.33, the restriction map S pAq Ñ S pB q
is surjective. Hence the inverse image of χ forms a non-empty extremal face
in the compact convex set S pAq. By the Krein-Milman theorem, there exists
an extremal point ϕ of S pAq in the inverse image of χ. Then ϕ is a pure
state which by construction satises ϕpa aq }a a}. It follows that the GNSrepresentation of ϕ is an irreducible representation. Moreover, }πϕ paqξϕ }2 ϕpa aq 0. Hence πϕ paq 0.
For commutative C -algebra, irreducible representations are easily classied.
Proposition 2.2.32. Let A be a unital commutative C -algebra. Then a
unital -representation π is irreducible if and only if π is one-dimensional.
In other words, all irreducible representations are characters.
Proof. Let π be an irreducible representation of A. Then for each a P A,
the operator π paq is itself an intertwiner of π . By Schur's lemma, π paq is
a scalar. As this holds for all a P A, it follows by irreducibility that π is
one-dimensional.
On the other hand, each one-dimensional representation is necessarily irreducible.
9 The actual Gelfand-Raikov theorem refers to the representation theory of locally com-
pact Hausdor groups, but as we will see later, it is a direct corollary of its C -algebra
version.
96
Chapter 2. Examples of Banach and C -algebras
We also have the following corollary concerning elds of C -algebras, saying
in a sense that elements in the global C -algebra are determined by their
values in the bers.
Corollary 2.2.33. Let X be a compact Hausdor space, and A a Cb pX qalgebra. Let a P A, and assume πx paq 0 for all x P X . Then a 0.
Proof. Let a P A be non-zero. By the Gelfand-Raikov theorem, we can nd
an irreducible representation π of A with π paq 0. As π is irreducible,
it follows that the self-intertwiner space of π is CidHπ . Hence π|Z pAq gives
a character on Z pAq. In particular, there exists x P X such that π pf q f pxqidHπ for all f P Cb pX q. We conclude that π factors through Ax . Hence
}πpaq} ¤ }πxpaq}, and πxpaq 0.
From Proposition 2.2.32, it follows that for A a commutative unital C algebra, the set SppAq can be identied with the set of all irreducible representations up to unitary equivalence. There is no obstruction to extend this
denition to the non-commutative case.
Denition 2.2.34. Let A be a unital C -algebra. We dene the set-theoretic
spectrum of A as
SppAq IrrpAq{ ,
the set of all irreducible representations of A modulo unitary equivalence.
Unlike for commutative C -algebras however, the spectrum SppAq can be
ill-behaved and retain very little of A itself. Nevertheless, SppAq can still be
equipped with a compact (but not necessarily Hausdor!) topology. This
topology is the one inherited from a cruder kind of spectrum.
Denition 2.2.35. Let A be a unital C -algebra. A 2-sided ideal J of A
is called primitive if J Kerpπq for some (non-zero) irreducible unital representation π. The primitive spectrum PrimpAq of A is the set of all
primitive ideals of A.
Remark 2.2.36. The primitive spectrum can be very small. Indeed, there
exist many simple C -algebras, whose only 2-sided ideals are t0u and the
whole C -algebra. For example, the Calkin algebra Q is simple.
2.2 The abstract Toeplitz algebra
97
Proposition 2.2.37. Let A be a unital C -algebra. For I a closed 2-sided
ideal of A, dene
VI
tJ P PrimpAq | I „ J u.
Then
tVI | I a 2-sided closed idealu
is the set of all closed subsets of a topology on PrimpAq.
The above topology on PrimpAq is referred to as the Jacobson topology.
V
Proof. Cleary Vt0u PrimpAq and VA H. For Iα a collection of closed
2-sided ideals, write I for the 2-sided ideal generated by all Iα . Then clearly
VI
XαVI
α
.
It remains to show that V is closed under nite unions.
Let I, J be two closed 2-sided ideals. Then clearly VI Y VJ
„ VI XJ .
Conversely, take P P VI XJ . Let π be an irreducible representation with
Kerpπ q P . Then the closure of π pJ qHπ is clearly an invariant subspace of
Hπ , since J is a two-sided ideal. By irreducibility, one has
either πpJ qHπ 0 and hence πpJ q 0, or
πpJ qHπ is dense in Hπ .
In the rst case, we obtain J „ P , hence P P VJ . In the second case,
we deduce from I X J „ P that π pxqπ py q 0 for all x P I, y P J , hence
π pI qπ pJ qHπ t0u. By density of π pJ qHπ in Hπ , this entails π pI q 0, hence
I „ P and P P VI . In both cases, we obtain P P VI Y VJ .
Hence VI Y VJ VI XJ .
Denition 2.2.38. The topological spectrum of A is the set SppAq equipped
with the topology inherited from the Jacobson topology from PrimpAq under
the map
Ker : SppAq Ñ PrimpAq,
rπs ÞÑ Kerpπq.
In other words, a subset of SppAq is open if and only if it is the inverse under
Ker of an open in PrimpAq.
98
Chapter 2. Examples of Banach and C -algebras
Remark 2.2.39. In general, the map SppAq
Ñ PrimpAq is not injective.
Indeed, one can show that for any innite-dimensional separable simple C algebra the set SppAq is not a singleton.
Proposition 2.2.40. The topological spaces SppAq and PrimpAq are compact.
Proof. As Ker : SppAq Ñ PrimpAq is surjective, it is open by denition of
the topology on SppAq. It hence suces to prove that PrimpAq is compact.
We use the following criterion for compactness: if Fα is any collection of
closed subsets for which the intersection of any nite subfamily is never
empty, then the intersection of all Fα is not empty.
For our situation, choose closed 2-sided ideals Iα and suppose that Xα VIα H. Then the closure of the C-algebra generated by all Iα contains A, and
in particular contains the unit of A. Now the C -algebra generated by the
Iα is just the closure of the linear span°of all Iα . Hence we can nd nitely
many°αi and xi P Iαi such that }1 i xi } 12 . In particular, we deduce
that i xi is invertible. But this means that the 2-sided ideal generated by
the Iαi is already A, hence Xi VIαi H.
When A is a commutative unital C -algebra, the space SppAq already carried
the topology of weak -convergence. Let us show that it coincides with the
topology inherited from PrimpAq. Note that for A commutative, one of course
has that the map SppAq Ñ PrimpAq is bijective.
Proposition 2.2.41. Let A be a commutative unital C -algebra. Then a net
of characters χα on A converges to a character χ for the weak -topology if
and only if Kerpχα q converges to Kerpχq for the Jacobson topology.
Proof. Suppose rst that χα Ñ χ for the weak -topology. Then by denition
we have Kerpχα q Ñ Kerpχq if and only if for any 2-sided closed ideal I of
A with I † Kerpχq, one has eventually I † Kerpχα q. But suppose that this
were not the case. Then we can nd a subnet χβ such that I „ Kerpχβ q
for all β . But then for each x P I , we have χpxq limβ χβ pxq 0. Hence
I „ Kerpχq, a contradiction.
Hence the map Ker : SppAq Ñ PrimpAq is a continuous bijection. To show
2.2 The abstract Toeplitz algebra
99
that it is a homeomorphism, it thus suces to prove that PrimpAq is Hausdor.
Now by the Gelfand-Neumark theorem, we may identify A Cb pSppAqq
(where SppAq has the weak -topology). Take χ, χ1 P SppAq, and let U and
U 1 be open neighborhoods of respectively χ and χ1 with U X U 1 H. Let
I be the closed 2-sided ideal of functions which vanish on U c , and similarly
I 1 the closed 2-sided ideal of functions which vanish on U 1c . Then VI „
PrimpAq consists of all Kerpχ2 q such that f pχ2 q 0 for all f P I . Hence,
by the Urysohn lemma, VIc is an open neighborhood of Kerpχq, and similarly
VIc1 is an open neighborhood of Kerpχ1 q. Clearly this description also gives
VIc X VIc1 H, as U X U 1 H. Hence PrimpAq is Hausdor.
As an example, let us examine the topological spectrum of the Toeplitz C algebra.
Lemma 2.2.42. Up to unitary equivalence, any irreducible -representation
of the Toeplitz C -algebra T is either the standard representation of T on
l2 pNq, or else a character
χz : T
Ñ C,
S
ÞÑ z
for some z P S1 . These representations are all mutually inequivalent.
Proof. The fact that the above representations are mutually inequivalent is
clear. It is also clear that all characters χz are irreducible.
Let us show that the standard representation T
In fact if ξ, η in l2 pNq, the rank one operator
Tξ,η : l2 pNq Ñ l2 pNq,
ζ
„ B pl2pNqq is irreducible.
ÞÑ xξ, ζ yη
lies in B0 pl2 pNqq „ T . It follows that any non-zero vector in l2 pNq is cyclic
for T , hence the representation of T on l2 pNq is irreducible.
It remains to show that the representations in the statement of the lemma
exhaust all irreducible representations up to unitary equivalence.
But let π be an irreducible representation of T . From the Wold decomposition appearing in Theorem 2.2.8, we may assume H l2 pX Nq ` G with
then π pS q SX ` u with u unitary. From the irreducibility of π , we conclude that either G t0u or X H. In the rst case, irreducibility implies
100
Chapter 2. Examples of Banach and C -algebras
that X must be a singleton (otherwise we have a direct sum decomposition
l2 pX Nq l2 pX1 Nq ` l2 pX2 Nq as representations for any decomposition X X1 \ X2 ), in which case this is just the ordinary representation
on l2 pNq. In the second case, π quotients to an irreducible representation of
Cb pS1 q, which is then necessarily a character. We deduce that in this case
π χz for some z P S1 .
Corollary 2.2.43. The map
Ker : SppT q Ñ PrimpT q
is bijective.
Proof. If π is the standard representation, then Kerpπq t0u. Hence Kerpπq Kerpχz q for any character z . Since clearly Kerpχz q Kerpχw q for z w, the
corollary follows.
From the above, we can write SppT q S1 Y tu, with representing the
standard representation on l2 pNq. It remains to determine the topology of
SppT q.
Proposition 2.2.44. A subset X „ SppT q is closed if and only if X SppT q
or X „ S1 is closed (for the ordinary topology on S1 ).
Proof. Under the identication SppT q PrimpT q, the point corresponds
to the zero ideal t0u. But clearly t0u P VI for a closed 2-sided ideal I if and
only if I t0u. It follows that SppAq is the only closed set containing .
It now suces to show that χzα Ñ χz in SppT q if and only if zα
This follows along similar lines as in Proposition 2.2.41.
Ñ z P S1.
One may hence visualize the spectrum of T as being the closed unit disc with
all points in the interior identied into a `fat open point' whose closure is the
whole unit disc.