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595

Documenta Math.

Hyperbolic Geometry
on Noncommutative Balls
Gelu Popescu1
Received: March 6, 2009
Revised: October 31, 2009
Communicated by Joachim Cuntz

Abstract. In this paper, we study the noncommutative balls
Cρ := {(X1 , . . . , Xn ) ∈ B(H)n : ωρ (X1 , . . . , Xn ) ≤ 1},

ρ ∈ (0, ∞],

where ωρ is the joint operator radius for n-tuples of bounded linear
operators on a Hilbert space. In particular, ω1 is the operator norm,
ω2 is the joint numerical radius, and ω∞ is the joint spectral radius.
We introduce a Harnack type equivalence relation on Cρ , ρ > 0, and
use it to define a hyperbolic distance δρ on the Harnack parts (equivalence classes) of Cρ . We prove that the open ball

[Cρ ] 0,

is the Harnack part containing 0 and obtain a concrete formula for the
hyperbolic distance, in terms of the reconstruction operator associated
with the right creation operators on the full Fock space with n generators. Moreover, we show that the δρ -topology and the usual operator
norm topology coincide on [Cρ ]l β

α\l β

ρ
0
otherwise.
Note that if {hβ }|β|≤q ⊂ H, then we have


+
∞ X
X
X
X

∗
k
hγ ⊗ e γ
hβ ⊗ e β  ,
Xα ⊗ r Rα˜  
h(ρI ⊗ I +
k=1 |α|=k

=ρ

X

|β|≤q

=ρ

X

|β|≤q

=ρ

X

|β|≤q

=

2

khβ k +

2

khβ k +

γ≥β; |β|,|γ|≤q

X

k=1 |α|=k

khβ k2 + +

X

|γ|≤q

|β|≤q

∞
X

X

*

X

|β|≤q

X

|α|≥1 |β|,|γ|≤q

X

Xα∗ hβ

γ>β; |β|,|γ|≤q

k

⊗ r Rα˜ eβ ,

X

|γ|≤q

hγ ⊗ e γ

r|α| heβα , eγ i hXα∗ hβ , hγ i

E
D
∗
h
,
h
r|γ\l β| Xγ\
γ
lβ β

hρKρ,X,r (γ, β)hβ , hγ i .

Documenta Mathematica 14 (2009) 595–651

+

612

Gelu Popescu

∗
Now, taking into account that Kρ,X,r (γ, β) = Kρ,X,r
(β, γ), we deduce relation
2
(2.2). Therefore, the condition Pρ (rA, R) ≤ c Pρ (rB, R), r ∈ [0, 1), implies

[Kρ,A,r (α, β)]|α|,|β|≤q ≤ c2 [Kρ,B,r (α, β)]|α|,|β|≤q

for any 0 < r < 1 and q = 0, 1, . . .. Taking r → 1 in the latter inequality, we
obtain item (iv).
Assume now that (iv) holds. Since c2 Kρ,B − Kρ,A is a positive semidefinite multi-Toeplitz kernel, due to Theorem 3.1 from [39] (see also the proof
of Theorem 5.2 from [47]), we find a completely positive linear map µ :
C ∗ (S1 , . . . , Sn ) → B(E) such that
µ(Sα ) = c2 Kρ,B (g0 , α) − Kρ,A (g0 , α) =

1 2
(c Bα − Aα )

ρ

2
for any α ∈ F+
n with |α| ≥ 1, and µ(I) = (c − 1)I. Since

P (rS, R) :=

∞ X
X

k=1 |α|=k

∗
rk Sα ⊗ Rα
˜ +I ⊗I +

∞ X
X

k=1 |α|=k

rk Sα∗ ⊗ Rα˜ ≥ 0

for r ∈ [0, 1), we deduce that
(µ ⊗ id)[P (rS, R)] =

∞ X
X
1 |α| 2 ∗
r [c Bα − A∗α ] ⊗ Rαe + (c2 − 1)I ⊗ I
ρ
k=1 |α|=k

∞ X
X
1 |α| 2
∗
+
r [c Bα − Aα ] ⊗ Rα

e
ρ
k=1 |α|=k

= c2 Pρ (rB, R) − Pρ (rA, R) ≥ 0,

which implies (ii).
Let us prove that (iv) =⇒ (vi). Assume that (iv) holds. Then we have Kρ,A ≤
c2 Kρ,B , where Kρ,X is the multi-Toeplitz kernel associated with X ∈ Cρ . Let
V := (V
W1 , . . . , Vn ) be the minimal isometric dilation of A := (A1 , . . . , An ). Then
KA = α∈F+
Vα H and ρPH Vα |H = Aα for any |α| ≥ 1. Similar properties hold
n
if W := (W1 , . . . , Wn ) is the minimal isometric dilation of B := (B1 , . . . , Bn ).
Hence, and taking into account that V1 , . . . , Vn and W1 , . . . , Wn are isometries
with orthogonal ranges, respectively, we have
‚
‚2
‚

‚
‚ X
‚
‚
‚ =
V
h
α
α
‚
‚
‚|α|≤m
‚
X
X
˙
¸
hhα , hα i +
Vα\l β hα , hβ +
=
|α|≤m

α>l β,|α|,|β|≤m

=

X

α>l β,|α|,|β|≤m

=

X

|α|≤m,|β|≤m

fi

1
Aα\l β hα , hβ
ρ

fl

+

X

|α|≤m

X

β>l α,|α|,|β|≤m

hhα , hα i +

˙ ∗
¸
Vβ\l α hα , hβ

X

β>l α,|α|,|β|≤m

D

hKρ,A (β, α)hα , hβ i = [Kρ,A (β, α)]|α|,|β|≤m hm , hm

Documenta Mathematica 14 (2009) 595–651

fi

E

1 ∗
A
hα , hβ
ρ β\l α

fl

Hyperbolic Geometry on Noncommutative Balls

613

for any m ∈ N and hm := ⊕|α|≤m hα ∈ ⊕|α|≤m Hα , where each Hα is a copy of
H. Similarly, we obtain
2


X
E
D



Wα hα

= [Kρ,B (β, α)]|α|,|β|≤m hm , hm .


|α|≤m

Taking into account that Kρ,A ≤ c2 Kρ,B , we deduce that






X
X







Wα hα
Vα hα
≤ c

.


|α|≤m

|α|≤m

Therefore, we can define an operator LB,A : KB → KA by setting


X
X
Vα hα
Wα hα  :=
(2.3)
LB,A 
|α|≤m

|α|≤m

for any m ∈ N and hα ∈ H, α ∈ F+
n . Note that LB,A is a bounded operator
with kLB,A k ≤ c. Since LB,A |H = IH , we have kLB,A k ≥ 1. It is easy to see
that LB,A Wi = Vi LB,A for i = 1, . . . , n. Therefore item (vi) holds.
Conversely, assume that there is an operator LB,A ∈ B(KB , KA ) with norm
kLB,A k ≤ c such that LB,A|H = IH and L
Wi = Vi LB,A , i = 1, . . . , n.
 B,AP
P
Then, we deduce that LB,A
|α|≤m Vα hα for any m ∈ N
|α|≤m Wα hα =

and hα ∈ H, α ∈ F+
n . The condition kLB,A k ≤ c implies

2

2
X

X





2


Vα hα
≤ c
Wα hα


,
|α|≤m

|α|≤m


which is equivalent to the inequality
D
E
D
E
[Kρ,A (β, α)]|α|,|β|≤m hm , hm ≤ c2 [Kρ,B (β, α)]|α|,|β|≤m hm , hm

for any m ∈ N and hm := ⊕|α|≤m hα ∈ ⊕|α|≤m Hα . Consequently, we deduce
item (iv). The proof is complete.

A closer look at the proof of Theorem 2.2 reveals that one can assume that
u(0) = I in part (iii), and one can also assume that ℜp(0) = I in the definition
H

of the Harnack domination A≺ B. We also remark that, due to Theorem 1.3,
H

we can add an equivalence to Theorem 2.2, namely, A≺ B if and only if
c

2

u(A1 , . . . , An ) + (ρ − 1)u(0) ≤ c [u(B1 , . . . , Bn ) + (ρ − 1)u(0)]
c
for any positive free pluriharmonic function u ∈ Harball
(B(E)).

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614

Gelu Popescu
H

Corollary 2.3. If A, B ∈ Cρ and A≺ B, then
H

kLB,Ak = inf{c > 1 : A≺ B}
c

= inf{c > 1 : Pρ (rA, R) ≤ c2 Pρ (rB, R)

for any

r ∈ [0, 1)}.

H

Moreover, A≺ B if and only if supr∈[0,1) kLrA,rB k < ∞. In this case,
kLA,B k = sup kLrA,rB k
r∈[0,1)

and the mapping r 7→ kLrA,rB k is increasing on [0, 1).
H

H

Proof. Assume that A≺ B. Then, due to Theorem 2.2, A≺ B if and only if there
c

is an operator LB,A ∈ B(KB , KA ) with kLB,Ak ≤ c such that LB,A |H = IH
and LB,A Wi = Vi LB,A for i = 1, . . . , n. Consequently, taking c = kLB,A k, we
H

deduce that A ≺

kLB,A k

B, which is equivalent to

Pρ (rA, R) ≤ kLB,Ak2 Pρ (rB, R)
H

for any r ∈ [0, 1). Hence, we have tA ≺

kLB,A k

tB for any t ∈ [0, 1). Applying again

Theorem 2.2 to the operators tA and tB, we deduce that kLtA,tB k ≤ kLB,A k.

Conversely, suppose that c := supr∈[0,1) kLrA,rB k < ∞. Since kLrA,rB k ≤ c,
H

Theorem 2.2 implies rA≺ rB for any r ∈ [0, 1) and, therefore, Pρ (rtA, R) ≤
c

2

H

c Pρ (rtB, R) for any t, r ∈ [0, 1). Hence, A≺ B and, consequently, kLB,A k ≤ c.
c

Therefore, kLA,B k = supr∈[0,1) kLrA,rB k. The fact that r 7→ kLrA,rB k is an
increasing function on [0, 1) follows from the latter relation. This completes
the proof.


We remark that if 1 ≤ m < n and u is a positive free pluriharmonic function
on [B(K)n ]1 , then the map
(X1 , . . . , Xm ) 7→ u(X1 , . . . , Xm , 0, . . . , 0)
is a positive free pluriharmonic function on [B(K)m ]1 . Moreover, if g is a
positive free pluriharmonic function on [B(K)m ]1 , then the map
(X1 , . . . , Xn ) 7→ g(X1 , . . . , Xm , 0, . . . , 0)
is a positive free pluriharmonic function on [B(K)n ]1 . Consequently, using
Corollary 1.2, one can easily deduce the following result.
Corollary 2.4. Let c > 0, ρ > 0, and 1 ≤ m < n. Consider two n-tuples
(A1 , . . . , Am ) ∈ B(H)m and (B1 , . . . , Bm ) ∈ B(H)m in the class Cρ and let
Documenta Mathematica 14 (2009) 595–651

Hyperbolic Geometry on Noncommutative Balls

615

(A1 , . . . , Am , 0, . . . , 0) and (B1 , . . . , Bm , 0, . . . , 0) be their extensions in B(H)n ,
H

respectively. Then (A1 , . . . , Am )≺ (B1 , . . . , Bm ) in Cρ ⊂ B(H)m if and only if
c

H

(A1 , . . . , Am , 0, . . . , 0)≺ (B1 , . . . , Bm , 0, . . . , 0)
c

in Cρ ⊂ B(H)n .

We recall (e.g. [43]) that if (T1 , . . . Tn ) is an n-tuple of operators,
then
the joint
P


spectral radius r(T1 , . . . , Tn ) < 1 if and only if limk→∞
|α|=k Tα Tα∗
= 0.
In what follows, we characterize the elements of Cρ which are Harnack dominated by 0.
H

Theorem 2.5. Let A := (A1 , . . . , An ) be in Cρ . Then A≺ 0 if and only if the
joint spectral radius r(A1 , . . . , An ) < 1.
Proof. Note that the map X 7→ Pρ (X, R) is a positive free pluriharmonic function on [B(H)n ]1 with coefficients in B(F 2 (Hn )) and has the factorization
(2.4)
Pρ (X, R) =
∗ −1
= (I − RX )−1 + (ρ − 2)I + (I − RX
)

∗ −1
∗
∗
= (I − RX
) [I − RX + (ρ − 2)(I − RX
)(I − RX ) + I − RX
] (I − RX )−1
∗ −1
∗
∗
= (I − RX
) [ρI + (1 − ρ)(RX
+ RX ) + (ρ − 2)RX
RX ] (I − RX )−1 ,

where RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the reconstruction operator associated with the n-tuple X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 . We remark that,
due to the fact that the spectral radius of RX is equal to the joint spectral radius r(X1 , . . . , Xn ), the factorization above holds for any X ∈ Cρ with
r(X1 , . . . , Xn ) < 1.
Now, using Theorem 2.2 part (ii) and the above-mentioned factorization, we
H

deduce that A≺ 0 if and only if there exists c > 0 such that

∗ −1
∗
∗
(I − RrA
) [ρI + (1 − ρ)(RrA
+ RrA ) + (ρ − 2)RrA
RrA ] (I − RrA )−1 ≤ ρc2 I

for any r ∈ [0, 1). Similar inequality holds if we replace the right creation
operators by the left creation operators. Then, applying the noncommutative
Poisson transform id ⊗ Peiθ R , where R := (R1 , . . . , Rn ), we obtain
(2.5)
∗
∗
∗
ρI +(1−ρ)(e−iθ RrA
+eiθ RrA )+(ρ−2)RrA
RrA ≤ ρc2 (I −re−iθ RA
)(I −reiθ RA )
for any r ∈ [0, 1) and θ ∈ R.
On the other hand, since A := (A1 , . . . , An ) ∈ Cρ , we have r(A1 , . . . , An ) ≤
1. Suppose that r(A1 , . . . , An ) = 1. Taking into account that r(RA ) =
r(A1 , . . . , An ), we can find λ0 ∈ T in the approximative spectrum of RA . Consequently, there is a sequence {hm } in H ⊗ F 2 (Hn ) such