Download Essential Normality and Boundary Representations

ON QUOTIENT MODULES OF H 2 (Dn ): ESSENTIAL NORMALITY AND
BOUNDARY REPRESENTATIONS
B. KRISHNA DAS, SUSHIL GORAI, AND JAYDEB SARKAR
Abstract. Let Dn be the open unit polydisc in Cn , n ≥ 1, and H 2 (Dn ) be the Hardy space
over Dn . We show that if θ ∈ H ∞ (Dn ) is an inner function, then the n-tuple of commuting
operators (Cz1 , . . . , Czn ) on the Beurling type quotient module Qθ is not essentially normal,
where
Qθ = H 2 (Dn )/θH 2 (Dn ), and Czj = PQθ Mzj |Qθ ,
j = 1, . . . , n, and n ≥ 3. We obtain also analogous characterizations for doubly commuting quotient modules of analytic Hilbert modules over C[z1 , . . . , zn ] and generalized Rudin’s
quotient modules of H 2 (D2 ).
Finally we give several results concerning boundary representations of C ∗ -algebras corresponding to a class of quotient modules of reproducing kernel Hilbert modules over C[z1 , . . . , zn ].
1. Introduction
Let H 2 (Dn ), n ≥ 1, denote the Hardy space of holomorphic functions on the unit polydisc
D = {z = (z1 , . . . , zn ) ∈ Cn : |zi | ≤ 1, i = 1, . . . , n}. That is
∑
∑
H 2 (Dn ) = {f =
ak z k ∈ O(Dn ) : ∥f ∥2 :=
|ak |2 < ∞},
n
k∈Nn
where N is the set of natural numbers including 0 and N = {k = (k1 , . . . , kn ) : kj ∈ N, j =
1, . . . , n} and z k = z1k1 · · · znkn . It is well known that H 2 (Dn ) is a reproducing kernel Hilbert
space corresponding to the Szegő kernel
n
∏
S(z, w) =
(1 − zi w̄i )−1 ,
(z, w ∈ Dn )
n
i=1
and (Mz1 , . . . , Mzn ) is a commuting tuple of isometries on H 2 (Dn ), where
(Mzi f )(w) = wi f (w).
(f ∈ H 2 (Dn ), w ∈ Dn , i = 1, . . . , n)
We represent the n-tuple of multiplication operators as a Hilbert module over C[z] :=
C[z1 , . . . , zn ] in the following sense:
(p, f ) ∈ C[z] × H 2 (Dn ) 7→ p(Mz1 , . . . , Mzn )f ∈ H 2 (Dn ).
We call H 2 (Dn ) as Hardy module over C[z1 , . . . , zn ] with the above module action. A closed
subspace S (Q) of H 2 (Dn ) is called a submodule (quotient module) if Mzi S ⊆ S for all
2000 Mathematics Subject Classification. 47A13, 47A20, 47L25, 47L40, 46L05, 46L06.
Key words and phrases. Hardy space, reproducing kernel Hilbert spaces, quotient modules, essentially
normality, boundary representations.
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2
DAS, GORAI, AND SARKAR
i = 1, . . . , n (Q⊥ ∼
= H 2 (Dn )/Q is a submodule). A quotient module Q is said to be of
Beurling type if
Q = Qθ := H 2 (Dn )/θH 2 (Dn ) ∼
= H 2 (Dn ) ⊖ θH 2 (Dn ),
for some inner function θ ∈ H ∞ (Dn ). We also use the notation Sθ to denote the submodule
θH 2 (Dn ).
A quotient module Q of H 2 (Dn ) is essentially normal if the cross commutators [CzQi , CzQ∗
]
j
is compact for all 1 ≤ i, j ≤ n, where
CzQi := PQ Mzi |Q
(i = 1, . . . , n)
is the module map corresponding to Q. When the quotient module Q is clear from the context
we shall write Czi instead of CzQi .
It is a well known consequence of a classical result of Beurling that if Q is a quotient module
of H 2 (D) then Q is of Beurling-type and essentially normal. This property, however, does not
hold in general: (1) Let n ≥ 2. Then Q is a Beurling type quotient module of H 2 (Dn ) if and
only if Q⊥ is a doubly commuting submodule (see [11]). (2) A Beurling type quotient module
Qθ ⊆ H 2 (D2 ) is essentially normal if and only if θ is a rational inner function of degree at
most (1, 1) (see [6]).
This paper is concerned with the essential normality of a class of quotient modules including
Beurling-type quotient modules of H 2 (Dn ), n ≥ 3. An added benefit of this consideration is
the study of boundary representations, in the sense of Arveson [1], [2], of algebras generated
by {IQ , Cz1 , . . . , Czn }.
We now recall the definition of a boundary representation. Let A be an operator space
or operator algebra with identity, and let C ∗ (A) be the C ∗ algebra generated by A. Then
an irreducible representation ω of C ∗ (A) is a boundary representation relative to A if ω|A
has a unique completely positive (CP) extension to C ∗ (A). It is important to note here that
we do not drop the irreducibility assumption of ω, which one usually do while constructing
enveloping C ∗ algebra of A ([5]). Let bdy(A) denotes the set of all boundary representations
of C ∗ (A) relative to A. One of the fundamental problems in operator algebras is to identify
A for which there are sufficiently many boundary representations in the following sense:
∩
ker ω = {0} and ∥(aij )∥ = sup ∥(ω(aij ))∥ ((aij ) ∈ Mm (A), m ∈ N).
ω∈bdy(A)
ω∈bdy(A)
If C ∗ (A) contains all the compact operators for an irreducible operator algebra A, then A
has sufficiently many boundary representation if and only if identity representation of C ∗ (A)
is a boundary representation relative to A [2].
Now let Q be an essentially normal quotient module of H 2 (Dn ), and let B(Q) := B(Cz1 , . . . , Czn )
and C ∗ (Q) := C ∗ (Cz1 , . . . , Czn ) be the Banach algebra and C ∗ -algebra generated by {IQ , Czi }ni=1 ,
respectively. It is now of interest to determine whether or not the identity is a boundary representation of C ∗ (Q) relative to B(Q). This problem has a complete solution for the case
n = 1 ([1],[2]):
Theorem 1.1 (Arveson). Let Qθ be a quotient module of H 2 (D). Then the identity representation of C ∗ (Qθ ) is a boundary representation relative to B(Qθ ) if and only if Zθ is a proper
ESSENTIAL NORMALITY AND BOUNDARY REPRESENTATIONS
3
subset of T, where Zθ consists of all points λ on T for which θ cannot be continued analytically
from D to λ.
Now let θ ∈ H ∞ (D2 ) be a rational inner function of degree at most (1, 1) and Qθ be the
corresponding Beurling type quotient module of H 2 (D2 ). Then the identity representation
of C ∗ (Qθ ) is a boundary representation relative to B(Qθ ) if and only if θ is a one variable
Blaschke product (see [6]).
In this paper, we study similar problems for certain classes of quotient modules of H 2 (Dn ),
n ≥ 2. Namely, we prove that the Beurling type quotient modules of H 2 (Dn ), n ≥ 3, are
not essential normal. Also we prove that the class of generalized Rudin quotient modules of
H 2 (D2 ) are not essentially normal. We will discuss these results in Chapter 3. In Section 4 we
restrict our study to a further special class of quotient modules and obtain some direct result
concerning boundary representations of C ∗ (Q). Moreover, for a quotient module Q of H 2 (Dn )
we obtain a sufficient condition, in terms of the joint essential spectrum of (Cz1 , . . . , Czn ), for
identity representation of C ∗ (Q) to be a boundary representation relative to B(Q). In the
section below we collect some definitions and elementary facts concerning quotient modules
of H 2 (Dn ).
2. Preparatory results
In this section we prove some elementary results and recall some definitions which will be
used in the later sections to prove our main results. For each w ∈ Dn , we denote by Kw the
normalized kernel function of H 2 (Dn ) and defined by
n
∏
√
1
1
Kw (z) :=
S(z, w) = (
(1 − |wi |2 ))
,
∥S(·, w)∥
1
−
w
z
i
i
i=1
(z ∈ Dn )
where (S(·, w))(z) = S(z, w) for all z ∈ Dn .
Lemma 2.1. Let l ∈ {1, . . . , n} be a fixed integer and wl = (w1 , . . . , wl−1 , wl+1 , . . . , wn ) be a
fixed point in Dn−1 . Then K(wl ,w) converges weakly to 0 as wj approaches to the boundary of
the unit disc, where (wl , w) = (w1 , . . . , wl−1 , w, wl+1 , . . . , wn ).
Proof. For each p ∈ C[z],
(2.1)
n
∏
√
√
2
⟨K(wl ,w) , p⟩ = p((wl , w)) 1 − |w|
1 − |wi |2 ,
i=1,i̸=j
which converges to zero as w approaches to the boundary of D. Now for an arbitrary f ∈
H 2 (Dn ), the result follows from the fact that ∥Kµ ∥ = 1 for all µ ∈ Dn and C[z] is dense in
H 2 (Dn ).
For a closed subspace S of a Hilbert space H, the orthogonal projection of H onto S will
be denoted by PS . Then by virtue of Beurling’s theorem it follow that
PSθ = Mθ Mθ∗ ,
PQθ = IH 2 (Dn ) − Mθ Mθ∗ ,
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DAS, GORAI, AND SARKAR
where θ ∈ H ∞ (Dn ) inner and Mθ is the multiplication operator defined by Mθ f = θf ,
f ∈ H 2 (Dn ). It also follows from the reproducing property of the Szegö kernel that
Mθ∗ K(·, w) = θ(w)K(·, w),
where K(·, w) := Kw , w ∈ Dn . In particular, one has
PSθ (Kw ) = Mθ Mθ∗ Kw = η(λ)ηKw .
(w ∈ Dn )
These observations yield the next lemma.
Lemma 2.2. Let Qθ be a quotient module of H 2 (Dn ) for some inner function θ. Then
(2.2)
PQθ (Kw ) = (1 − θ(w)θ)Kw .
(w ∈ Dn )
We now recall the notion of analytic Hilbert modules over C[z] [3]. Let k : D × D → C
be a positive definite function such that k(z, w) is analytic in z and co-analytic in w. Let
Hk ⊆ O(D, C) denote the corresponding reproducing kernel Hilbert space. The Hilbert space
Hk is said to be reproducing kernel Hilbert module over C[z] if the multiplication operator Mz
is bounded on Hk .
Definition 2.3. A reproducing kernel Hilbert module Hk over C[z] is said to be an analytic
Hilbert module over C[z] if k −1 (z, w) is a polynomial in z and w̄.
Typical examples of analytic Hilbert modules are the Hardy module H 2 (D) and the weighted
Bergman modules over D.
It is known that a quotient module of an analytic Hilbert module is irreducible, that is,
Cz does not have any non-trivial reducing subspace (see Theorem 3.3 and Lemma 3.4 in [3]).
With all of this in mind, the next result shows that C ∗ (Q) is commutative if and only if
Q∼
= C.
Lemma 2.4. Let Q be a non-zero quotient module of a analytic Hilbert module H over C[z].
Then [Cz , Cz∗ ] = 0 if and only if Q is one dimensional.
Proof. First note that for any quotient module Q of H, the C ∗ algebra C ∗ (Q) is irreducible.
If Cz is normal then by Fuglede’s theorem Cz ∈ C ∗ (Q)′ = CI. Thus C ∗ (Q) = CI and,
therefore, Q is one dimensional. The converse part is trivial and the proof follows.
In the sequel, we will often identify Mzi on H 2 (Dn ) with IH 2 (D) ⊗ · · · ⊗ Mz ⊗ · · · ⊗ IH 2 (D) ,
|{z}
i-th place
i = 1, . . . , n, on H 2 (D) ⊗ · · · ⊗ H 2 (D), the n-fold Hilbert space tensor product of the Hardy
module.
We end this section with a result on essential normality of a Beurling type quotient module
Qθ , where θ is a one variable inner function.
Lemma 2.5. Let θ ∈ H ∞ (Dn ) be a one variable inner function and n ≥ 3. Then Qθ is not
essentially normal.
ESSENTIAL NORMALITY AND BOUNDARY REPRESENTATIONS
5
Proof. Without loss of generality we may assume that θ(z) = θ′ (z1 ) for some inner function
θ′ ∈ H ∞ (D). It then follows that Sθ = Sθ′ ⊗ H 2 (Dn−1 ) and hence
Qθ = H 2 (Dn ) ⊖ θH 2 (Dn ) = Qθ′ ⊗ H 2 (Dn−1 ).
Now we compute the self commutator of Cz2 :
[Cz2 , Cz∗2 ] = PQθ Mz2 Mz∗2 |Qθ − PQθ Mz∗2 PQθ Mz2 |Qθ
= PQθ Mz2 Mz∗2 |Qθ − IQθ + PQθ Mz∗2 PSθ Mz2 |Qθ .
Now from the fact that
PSθ Mz2 |Qθ′ ⊗C⊗H 2 (Dn−1 ) = PSθ′ ⊗ IH 2 (D) ⊗ IH 2 (Dn−1 ) Mz2 |Qθ′ ⊗C⊗H 2 (Dn−1 ) = 0,
and
Mz∗2 |Qθ′ ⊗C⊗H 2 (Dn−1 ) = 0,
we conclude that
[Cz2 , Cz∗2 ]|Qθ′ ⊗C⊗H 2 (Dn−1 ) = −IQθ |Qθ′ ⊗C⊗H 2 (Dn−1 ) = −IQθ′ ⊗C⊗H 2 (Dn−2 ) .
Since n ≥ 3, this implies [Cz2 , Cz∗2 ]|Qθ′ ⊗C⊗H 2 (Dn−1 ) is not compact and hence the commutator
[Cz2 , Cz∗2 ] is not compact. This completes the proof of the lemma.
3. Essential normality
Our purpose in this section is to prove the a list of results concerning essential normality
for a class of quotient modules. We start with the class of Beurling type quotient modules of
H 2 (Dn ), n ≥ 3.
Theorem 3.1. Let θ be an inner function in H ∞ (Dn ) and n ≥ 3. Then Qθ is not essentially
normal.
Proof. By Lemma 2.5 one may assume without loss of generality that θ depends on both z1
and z2 variables. We now show that [Cz1 , Cz∗2 ] is not compact. To see this, we compute
[Cz1 , Cz∗2 ] = PQθ Mz1 Mz∗2 |Qθ − PQθ Mz∗2 PQθ Mz1 |Qθ = PQθ Mz∗2 PSθ Mz1 |Qθ
= PQθ Mz∗2 PSθ ⊖(z1 Sθ +z2 Sθ ) Mz1 |Qθ + PQθ Mz∗2 Pz1 Sθ +z2 Sθ Mz1 |Qθ .
But since Mz1 and Mz2 are isometries, we also have
Pz1 Sθ Mz1 PQθ = 0, and PQθ Mz∗2 Pz2 Sθ = 0,
and so
PQθ Mz∗2 Pz1 Sθ +z2 Sθ Mz1 |Qθ = 0,
and hence
[Cz1 , Cz∗2 ] = PQθ Mz∗2 PSθ ⊖(z1 Sθ +z2 Sθ ) Mz1 |Qθ .
On the other hand, since Sθ = θH 2 (Dn ), we must have that
Sθ ⊖ (z1 Sθ + z2 Sθ ) = θ(C ⊗ C ⊗ H 2 (Dn−2 ),
and hence Mz∗2 (Sθ ⊖ (z1 Sθ + z2 Sθ )) ⊆ Qθ . Consequently
[Cz1 , Cz∗2 ] = Mz∗2 PSθ ⊖(z1 Sθ +z2 Sθ ) Mz1 |Qθ .
6
DAS, GORAI, AND SARKAR
We now note that by virtue of Lemma 2.1, the assertion of the theorem will follow, if we show
that ⟨[Cz1 , Cz∗2 ]Kw , Kw ⟩ does not approaches to 0 as wj approached to ∂D, for some fixed
3 ≤ j ≤ n, and all other co-ordinates of w = (w1 , . . . wj−1 , wj , wj+1 , . . . , wn ) ∈ Dn remain
fixed.
To this end, let w ∈ Dn . Then since {θz3m3 · · · znmn : m3 , . . . , mn ∈ N} is an orthonormal basis
of Sθ ⊖ (z1 Sθ + z2 Sθ ), we have
PSθ ⊖(z1 Sθ +z2 Sθ ) (z2 Kw ) =
∑
⟨z2 Kw , θz3m3 · · · znmn ⟩θz3m3 · · · znmn
m3 ,...,mn ∈N
=
∑
⟨Kw , z3m3 · · · znmn (Mz∗2 θ)⟩θz3m3 · · · znmn
m3 ,...,mn ∈N
=
∑
1
θ
∥S(·, w)∥ m ,...,m
3
=
Mz∗2 θ(w)
2
∏
(w3 z3 )m3 . . . (wn zn )mn Mz∗2 θ(w)
n ∈N
(1 − |wj | )
2
1
2
j=1
n
(∏
)
Kwi θ.
i=3
Then
⟨[Cz1 , Cz∗2 ]Kw , Kw ⟩ = ⟨Mz∗2 PSθ ⊖(z1 Sθ +z2 Sθ ) Mz1 PQθ Kw , Kw ⟩
= ⟨Mz1 PQθ Kw , PSθ ⊖(z1 Sθ +z2 Sθ ) (z2 Kw )⟩
2
n
⟨
⟩
∏
∏
1
Kw i θ
= Mz∗2 θ(w) (1 − |wj |2 ) 2 Mz1 PQθ Kw ,
j=1
=
Mz∗2 θ(w)
2
∏
i=3
(1 − |wj | )
2
1
2
⟨
Mz1 (1 − θ(w)θ)Kw ,
n
∏
⟩
Kw i θ ,
i=3
j=1
∏
where the last equality follows from (2.2). Since Mz∗1 ( ni=3 Kwi ) = 0 and Mθ∗ Mθ = IH 2 (Dn ) ,
we have
⟨Mz1 θKw ,
n
∏
i=3
Therefore
⟩
Kwi θ = ⟨θMz1 Kw ,
n
∏
i=3
⟩
Kw i θ =
n
∏
⟨Kw , Mz∗1 (
i=3
⟩
Kwi ) = 0.
ESSENTIAL NORMALITY AND BOUNDARY REPRESENTATIONS
⟨[Cz1 , Cz∗2 ]Kw , Kw ⟩
=
Mz∗2 θ(w)
2
∏
(1 − |wj | )
2
1
2
⟨
Mz1 Kw ,
j=1
=
Mz∗2 θ(w)
2
∏
2
∏
(1 − |wj | )
2
2
Kw ,
n
∏
=
⟩
Kwi (Mz∗1 θ)
i=3
1
(1 − |wj |2 ) 2
(
Mz∗1 θ(w)
2
∏
)
∏
1
1
∥S(·, w)∥ i=3 (1 − |wj |2 ) 12
n
Mz∗1 θ(w)
j=1
Mz∗2 θ(w)
⟩
Kw i θ
i=3
⟨
1
j=1
= Mz∗2 θ(w)
n
∏
7
(1 − |wj |2 ).
j=1
But θ depends on both z1 and z2 variables, hence
Mz∗1 θ ̸= 0 ̸= Mz∗2 θ,
and it follows that there exists a l ∈ {3, . . . , n} such that the limit of
Mz∗2 θ(w)
Mz∗1 θ(w)
2
∏
(1 − |wj |2 ),
j=1
as wl approaches to ∂D keeping all other coordinates of w fixed, is a non-zero number. This
completes the proof.
We now proceed to the case of doubly commuting quotient modules of a large class of
Hilbert module over C[z1 , . . . , zn ].
Let {ki }ni=1 be n positive definite functions on D. Then HK := HK1 ⊗ · · · ⊗ HKn is said
to be an analytic Hilbert module ∏
over C[z] if Hki is analytic for all i = 1, . . . , n. In this case,
n
HK ⊆ O(D , C) and K(z, w) = ni=1 ki (zi , wi ) is the reproducing kernel function of HK (see
[3]).
Now let Q be a quotient module of an analytic Hilbert module HK over C[z]. It is known
that Q is doubly commuting (that is, [Czi , Cz∗j ] = 0 for all 1 ≤ i < j ≤ n) if and only if
Q = Q1 ⊗ · · · ⊗ Qn for some quotient module Qi of Hki , i = 1, . . . , n (see [7], [3] and [9]).
Theorem 3.2. Let Q = Q1 ⊗ · · · ⊗ Qn be a doubly commuting quotient module of an analytic
Hilbert module HK = Hk1 ⊗ · · · ⊗ Hkn over C[z], n ≥ 2. Then Q is essentially normal if and
only if one of the following holds:
(i) Q is finite dimensional.
(ii) There exits i ∈ {1, . . . , n} such that Qi is an infinite dimensional essentially normal
quotient module of Hki and Qj ∼
= C for all j ̸= i.
Proof. Let Q = Q1 ⊗ · · · ⊗ Qn be an infinite dimensional essentially normal quotient module.
Then at least one of the Qi , 1 ≤ i ≤ n, is infinite dimensional. Without loss of any generality
8
DAS, GORAI, AND SARKAR
we assume that Qn is infinite dimensional. For all i = 1, . . . , n, we now compute the selfcommutators:
[Czi , Cz∗i ] = PQ Mzi Mz∗i |Q − PQ Mz∗i PQ Mzi |Q
= PQ1 ⊗ · · · ⊗ PQi−1 ⊗ [Cz , Cz∗ ]i ⊗PQi+1 ⊗ · · · ⊗ PQn ,
| {z }
(3.3)
i-th place
[Cz , Cz∗ ]i
where
is the self-commutator of the quotient module Qi . Since Qn is infinite dimensional, the compactness of [Czi , Cz∗i ] implies that [Si , Si∗ ] = 0 for all 1 ≤ i ≤ n − 1. Therefore,
by Lemma 2.4, it follows that Qi ∼
= C, i = 1, . . . , n − 1.
Finally, for i = n, the compactness of [Czn , Cz∗n ] = PQ1 ⊗ · · · ⊗ PQn−1 ⊗ [Cz , Cz∗ ]n implies that
[Cz , Cz∗ ]n is compact, that is, Qn is essentially normal. For the sufficient part, it is enough
to show that (ii) implies Q is essentially normal. Again, without loss of generality assume
that Qn is infinite dimensional essentially normal quotient module. Then it readily follows
from (3.3) that [Czi , Cz∗i ] = 0, i = 1, . . . , n − 1, and [Czn , Cz∗n ] is compact. Now the proof
follows from Fuglede-Putnam Theorem.
We recall that all quotient modules of the Hardy modules H 2 (D) are essentially normal.
Hence, in the particular case of H 2 (Dn ), n ≥ 2, we obtain the following result.
Corollary 3.3. Let Q = Q1 ⊗ · · · ⊗ Qn be a doubly commuting quotient module of H 2 (Dn ),
n ≥ 2. Then Q is essentially normal if and only if one of the following holds:
(i) Q is finite dimensional.
(ii) There exits i ∈ {1, . . . , n} such that Qi is infinite dimensional and Qj ∼
= C for all
j ̸= i.
We now restrict our attention to n = 2 case and turn to formulate the definition of a
2
generalized Rudin quotient module of H 2 (D2 ). Let Ψ = {ψn }∞
n=0 ⊆ H (D) be a sequence of
2
increasing finite Blaschke products and Φ = {φn }∞
n=0 ⊆ H (D) be a sequence of decreasing
Blaschke products, that is, ψn+1 /ψn and φn /φn+1 are non-constant inner functions for all
n ≥ 0. Then the generalized Rudin quotient module corresponding to the inner sequences Ψ
and Φ is denoted by QΨ,Φ , and defined by
∞
∨
(
)
QΨ,Φ :=
Qψn ⊗ Qφn .
n=0
The submodule corresponding to QΨ,Φ is denoted by SΨ,Φ := H 2 (D2 ) ⊖ QΨ,Φ . The following
representations of QΨ,Φ and SΨ,Φ are very useful:
⊕
⊕
(3.4) QΨ,Φ =
(Qψn ⊖ Qψn−1 ) ⊗ Qφn and SΨ,Φ = Q′ ⊗ H 2 (D)
(Qψn ⊖ Qψn−1 ) ⊗ Sφn ,
n≥0
n≥0
′
where Qψ−1 := {0} and Q = H (D) ⊖ ∨n≥0 Qψn .
We now consider the issue of essential normality of a Rudin quotient module.
2
Theorem 3.4. Let QΨ,Φ be as above for an increasing sequence of finite Blaschke products
Ψ = {ψn }n≥0 and a decreasing sequence of Blaschke products Φ = {φn }n≥0 . Then QΨ,Φ is not
essentially normal.
ESSENTIAL NORMALITY AND BOUNDARY REPRESENTATIONS
9
Proof. Let bβ , the Blaschke factor corresponding to β ∈ D, be a factor of ψm+1 /ψm for some
m ≥ 0. For contradiction, we assume that QΨ,Φ is essentially normal. Then, as ψm is a finite
Blaschke product, [Cψm (z1 ) , Cψ∗ m (z1 ) ] is compact, where Cψm (z1 ) = PQ Mψm (z1 ) |Q and Q := QΨ,Φ .
Now letting S := SΨ,Φ , we have
[Cψm (z1 ) , Cψ∗ m (z1 ) ] = PQ Mψm (z1 ) Mψ∗m (z1 ) |Q − PQ Mψ∗m (z1 ) PQ Mψm (z1 ) |Q
= −PQ (I − Mψm (z1 ) Mψ∗m (z1 ) )|Q + PQ Mψ∗m (z1 ) PS Mψm (z1 ) |Q
(3.5)
= −PQ (PQψm ⊗ I)|Q + PQ Mψ∗m (z1 ) PS Mψm (z1 ) |Q .
Since φm+1 is an infinite Blaschke products, there is a sequence of {λi } in D such that
Kλi ∈ Qφm+1 and λi approaches to ∂D as i → ∞. Furthermore, since Kβ ⊗ Kλi ∈ Q and
ψm Kβ ⊗ Kλi ∈ (Qψm+1 ⊖ Qψm ) ⊗ Qφm+1 , we have PS (ψm Kβ ⊗ Kλi ) = 0 (by (3.4)), and hence
PQ Mψ∗m (z1 ) PS Mψm (z1 ) (Kβ ⊗ Kλi ) = 0.
Finally from (3.5), we have
⟨[Cψm (z1 ) , Cψ∗ m (z1 ) ](Kβ ⊗ Kλi ), Kβ ⊗ Kλi ⟩ = −⟨PQψm (Kβ ) ⊗ Kλi , Kβ ⊗ Kλi ⟩
= −⟨(1 − ψm (β)ψm )Kβ , Kβ ⟩
= −(1 − |ψm (β)|2 ),
which does not converges to 0 as λi approaches to ∂D, and we have the desired contradiction.
This completes the proof.
4. Boundary Representations
In this section, we study boundary representations for a class of quotient modules of analytic
Hilbert modules over C[z].
First we prove a general result for minimal tensor products of C ∗ -algebras. For two vector
spaces V1 and V2 we denote by V1 ⊗V2 the algebraic tensor product of vector spaces. We
denote the minimal tensor product of two C ∗ -algebras A1 and A2 by A1 ⊗ A2 .
Theorem 4.1. Let Ai be an unital subalgebra of B(Hi ) for some Hilbert space Hi , and let
C ∗ (Ai ) be the irreducible C ∗ -algebras generated by Ai in B(Hi ), i = 1, 2. Set A := (A1 ⊗A2 ),
the norm closure of A1 ⊗A2 in B(H1 ⊗ H2 ). Then the following are equivalent
(i) idC ∗ (A1 )⊗C ∗ (A2 ) is a boundary representation for C ∗ (A1 ) ⊗ C ∗ (A2 ) relative to A.
(ii) idC ∗ (Ai ) is a boundary representation for C ∗ (Ai ) relative to Ai for all i = 1, 2.
Proof. Suppose (i) holds and, for contradiction, we assume that idC ∗ (A1 ) is not a boundary
representation of C ∗ (A1 ) relative to A1 . Then there exists a CP map τ : C ∗ (A1 ) → B(H1 )
different from idC ∗ (A1 ) , but τ = idC ∗ (A1 ) on A. Then the CP map τ ⊗ idC ∗ (A2 ) : C ∗ (A1 ) ⊗
C ∗ (A2 ) → B(H1 ⊗ H2 ) is a CP extension of the map idC ∗ (A1 )⊗C ∗ (A2 ) |A to C ∗ (A1 ) ⊗ C ∗ (A2 )
and τ ⊗ idC ∗ (A2 ) ̸= idC ∗ (A1 )⊗C ∗ (A2 ) . This is a contradiction.
On the other hand suppose (ii) holds. It follows from [1, Theorem 2.2.7]
∨ that idC ∗ (A1 )⊗C ∗ (A2 ) is
a boundary representation relative to the linear subspace I ⊗ C ∗ (A1 ) C ∗ (A1 ) ⊗ I. Thus it is
10
DAS, GORAI, AND SARKAR
enough to show that any CP extension
τ : C ∗ (A1 )⊗C ∗ (A2 ) → B(H1 ⊗H2 ) of idC ∗ (A1 )⊗C ∗ (A2 ) |A
∨
agrees on the subspace I ⊗ C ∗ (A1 ) C ∗ (A1 ) ⊗ I, that is,
τ (I ⊗ T2 ) = I ⊗ T2 and τ (T1 ⊗ I) = T1 ⊗ I.
(T1 ∈ C ∗ (A1 ), T2 ∈ C ∗ (A2 ))
To this end, let ω on B(H1 ) be a positive linear functional and
(ω ⊗ idB(H2 ) ) ◦ τ : C ∗ (A1 ) ⊗ C ∗ (A2 ) → B(H2 )
be the corresponding CP map. By identifying I ⊗ C ∗ (A2 ) with C ∗ (A2 ), one can see that the
CP map (ω ⊗ idB(H2 ) ) ◦ τ |I⊗C ∗ (A2 ) is an extension of ω(I) idC ∗ (A2 ) |A2 . Then by the assumption
we have
(ω ⊗ idB(H2 ) ) ◦ τ (I ⊗ T ) = ω(I)T = ω ⊗ idC ∗ (A2 ) (I ⊗ T ) (T ∈ C ∗ (A2 )).
By linearity, the above equality is also true for any linear functional ω on B(H1 ), and this
suggests that τ = idC ∗ (A1 )⊗C ∗ (A2 ) on I ⊗ C ∗ (A2 ). Similarly, by considering linear functionals
on B(H2 ) and repeating the above arguments, one can show that τ = idC ∗ (A1 )⊗C ∗ (A2 ) on
C ∗ (A1 ) ⊗ I. This completes the proof.
Remark 4.2. By using the same techniques one can generalize the above result for finite
number of irreducible generating C ∗ -algebras C ∗ (Ai ) corresponding to unital algebras Ai of
B(Hi ), i = 1, . . . , n.
As a straightforward consequence of Remark 4.2 we have the following:
Theorem 4.3. Let Q = Q1 ⊗ · · · ⊗ Qn be a doubly commuting quotient module of an analytic
Hilbert module H = HK1 ⊗ · · · ⊗ HKn over C[z]. Then the following are equivalent:
(i) The identity representation of C ∗ (Q) is a boundary representation relative to B(Q).
(ii) The identity representation of C ∗ (Qi ) is a boundary representation relative to B(Qi ),
for all all i = 1, . . . , n.
Proof. The result follows from Remark 4.2 and the fact that
C ∗ (Q) = C ∗ (Q1 ) ⊗ · · · ⊗ C ∗ (Qn ),
and
B(Q) = B(Q1 )⊗ · · · ⊗B(Qn ),
where the closure is under the norm topology of B(Q).
The following result is now an immediate consequence of Theorem 4.3 and Theorem 1.1.
Theorem 4.4. Let Q = Qθ1 ⊗ · · · ⊗ Qθn be a doubly commuting quotient module of H 2 (Dn )
for some one variable inner function ηi ’s, i = 1, . . . , n. Then the following are equivalent.
(i) The identity representation of C ∗ (Q) is a boundary representation relative to B(Q).
(ii) The identity representation of C ∗ (Qθi ) is a boundary representation relative to B(Qθi )
for all i = 1, . . . , n.
(iii) Zθi is a proper subset of T for all i = 1, . . . , n, where Zθi consists of all points λ on T
for which θi cannot be continued analytically from D to λ.
ESSENTIAL NORMALITY AND BOUNDARY REPRESENTATIONS
11
Acknowledgment: The first two authors are grateful to Indian Statistical Institute, Bangalore Centre for warm hospitality. The first named author also thanks NBHM for financial
support. The second named author is supported by an INSPIRE faculty fellowship (IFA-MA02) funded by DST.
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Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road,
Bangalore, 560059, India
E-mail address: [email protected], [email protected]
Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road,
Bangalore, 560059, India
E-mail address: [email protected]
Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road,
Bangalore, 560059, India
E-mail address: [email protected], [email protected]