Hyperbolic geometry on noncommutative polyballs

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by  Gelu Popescu

HYPERBOLIC GEOMETRY ON NONCOMMUTATIVE POLYBALLS
GELUPOPESCU
Abstract. This paper is an introduction to the hyperbolic geometry of noncommutative polyballs
7 Bn(H) in B(H)n1+···+nk, where n = (n1,...,nk) ∈ Nk and B(H) is the algebra of all bounded
1 linear operators on a Hilbert space H. We use the theory of free pluriharmonicfunctions on polyballs
0
and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type
2 metricsonBn(H),studytheirproperties,andobtainhyperbolicversionsofSchwarz-Picklemmaforfree

n holomorphicfunctionsonpolyballs. Asaconsequence, thepolyballscanbeviewedasnoncommutative
a hyperbolic spaces. When specialized to the regular polydisk Dk(H) (which corresponds to the case
J n1 = ··· = nk = 1), our hyperbolic metric δH is complete and invariant under the group Aut(Dk)
3 of allfreeholomorphicautomorphisms ofDk(H), and theδH-topology inducedonDk(H)isthe usual
operatornormtopology. TherestrictionofδH tothescalarpolydiskDk isequivalenttotheKobayashi
] distance on Dk. Mostof the resultsof this paper arepresented inthe moregeneral setting ofHarnack
A (resp.Poisson)partsoftheclosedpolyballBn(H)−.
F
.
h
t
a Contents
m
[ Introduction

1 1. Harnack and Poisson equivalence relations on the closed polyball
v 2. Hyperbolic metric on the Harnack parts of the closed polyball
6 3. A metric on the Poisson parts of the closed polyball
6
4. Hyperbolic metric on the regular polydisk
7
0 References
0
.
1 Introduction
0
7 A theory of free holomorphic functions on noncommutative polydomains which admit universal op-
1
erator models has been developed in [18], [19], [23], [24], [26], and [28]. These results played a cru-
:
v cial role in our work on the curvature invariant [25], the Euler characteristic [27], and the group of
Xi free holomorphic automorphisms on noncommutative regular polyballs [28]. The regular polyball Bn,
r n = (n1,...,nk) ∈ Nk, is a noncommutative analogue of the scalar polyball (Cn1)1 ×···× (Cnk)1
a and has a universal model S := {Si,j} consisting of left creation operators acting on the tensor product
F2(H )⊗···⊗F2(H ) offull Fockspaces. As aconsequence,the theoryoffree holomorphicfunctions
n1 nk
onnoncommutative polydomains is related, via noncommutative Berezin transforms,to the study of the
operator algebras generated by the universal models associated with the polydomains, as well as to the
theory of functions in several complex variable([13], [30], [31]). We remark that, in general, one can
view the free holomorphic functions on noncommutative polydomains as noncommutative functions in
the sense of [10].
Recently [29], we obtained structure theorems characterizing the bounded (resp. positive) free k-
pluriharmonic functions on regular polyballs. These results will play an important role in the present
paper which is an introduction to the hyperbolic geometry of noncommutative polyballs. The main goal
is to introduce hyperbolic type metrics on these polyballs, study their basic properties, and provide an
Date:October30,2016.
2000 Mathematics Subject Classification. Primary: 47A56; 51M10Secondary: 46L52; 47A13.
Keywordsandphrases. Noncommutativepolyball;Hyperbolicgeometry;Poincar´edistance;Berezintransform;Poisson
kernel;Fockspace;Pluriharmonicfunction;Freeholomorphicfunction;Schwarz-Picklemma.
ResearchsupportedinpartbyNSFgrantDMS1500922.
1
2 GELUPOPESCU
analogue of Schwarz-Pick lemma in this setting. As a consequence, the regular polyballs can be viewed
as noncommutative hyperbolic spaces.
Poincar´e’s discovery of a conformally invariant metric on the open unit disc D := {z ∈ C : |z| < 1}
of the complex plane is at the heart of geometric function theory. The hyperbolic (Poincar´e) distance is
defined on D by
1 1+|ϕ (w)|
δ (z,w):= ln z , z,w∈D,
P 2 1−|ϕ (w)|
z
where ϕ is the automorphism of D given by ϕ (w) = w−z , and it is invariant under the conformal
z z 1−z¯w
automorphisms of D, i.e.
δ (ϕ(z),ϕ(w))=δ (z,w), z,w∈D,
P P
for all ϕ ∈ Aut(D). Moreover, (D,δ ) is a complete metric space and the δ -topology induced on the
P P
opendiskistheusualplanartopology. Schwarz-Picklemmaassertsthatanyanalyticfunctionf :D→D
is distance-decreasing with respect to δ , i.e.
P
δ (f(z),f(w))≤δ (z,w), z,w∈D.
P P
This result has had profound implications in the development of geometric function theory. It has been
generalized to higher dimensional complex spaces in various ways (see [11], [12]). Bergman (see [3])
introduced an analogue of the Poincar´e distance for the open unit ball B := {z ∈ Cm : kzk < 1},
m 2
which has properties similar to those of δ . There is a large literature concerning invariant metrics,
P
hyperbolic manifolds, and the geometric viewpoint of complex function theory (see [11], [12], [9], [36],
and[14]andthe referencesthere in). There areseveralextensionsofthe Poincar´e-Bergmandistanceand
related topics to more general domains. We mention the work of by L. Harris ([6], [7], [8]) in the setting
of JB∗-algebras (see also the book by H. Upmeier [35]), and the work of I. Suciu ([32], [33]), Foia¸s ([5]),
andAndˆo-Suciu-Timotin([1])onHarnackpartsofcontractionsandHarnacktypedistancesbetweentwo
contractions on Hilbert spaces. In [21] (see also [20], [22]), we introduced a hyperbolic metric on the
noncommutative ball
[B(H)m] := (X ,...,X )∈B(H)m : kX X∗+···+X X∗k1/2 <1 , m∈N,
1 1 m 1 1 m m
n o
where B(H) denotes the algebra of all bounded linear operators on a Hilbert space H, which is a non-
commutativeextensionofthe Poincar´e-Bergmanmetric onthe openunit ballofCm. We alsoobtaineda
Schwarz-Picklemma for free holomorphic functions on [B(H)m] with respect to the hyperbolic metric.
1
To present our results, we introduce the regular polyballs. We denote by B(H)n1 ×c···×cB(H)nk,
whereni ∈N:={1,2,...},thesetofalltuplesX:=(X1,...,Xk)inB(H)n1×···×B(H)nk withtheprop-
ertythat the entries of X :=(X ,...,X ) arecommuting with the entries ofX :=(X ,...,X )
s s,1 s,ns t t,1 t,nt
for any s,t ∈ {1,...,k}, s 6= t. Note that the operators X ,...,X are not necessarily commuting.
s,1 s,ns
Let n:=(n ,...,n ) and define the polyball
1 k
Pn(H):=[B(H)n1]1×c···×c[B(H)nk]1.
If A is a positive invertible operator, we write A > 0. The regular polyball on the Hilbert space H is
defined by
Bn(H):={X∈Pn(H): ∆X(I)>0},
where the defect mapping ∆X :B(H)→B(H) is given by
∆X :=(id−ΦX1)◦···◦(id−ΦXk),
and Φ :B(H)→B(H) is the completely positive linear map defined by
Xi
ni
Φ (Y):= X YX∗ , Y ∈B(H).
Xi i,j i,j
j=1
X
Note that if k = 1, then Bn(H) coincides with the noncommutative unit ball [B(H)n1]1. We remark
that the scalar representation of the (abstract) regular polyball Bn :={Bn(H): H is a Hilbert space} is
Bn(C) = Pn(C) = (Cn1)1×···×(Cnk)1. Extending Poincar´e’s result [13] that the open ball of Cm is
not biholomorphic equivalent to the polydisk Dk, k ≥2, we provedin [28] that the noncommutative ball
[B(H)m]1 isnotfreebiholomorphicequivalenttotheregularpolyballBn(H),wheren=(n1,...,nk)and
HYPERBOLIC GEOMETRY ON NONCOMMUTATIVE POLYBALLS 3
k≥2. As in the classicalcase ([30], [31], [13]), one expects significantdifferences between the hyperbolic
geometry of the ball [B(H)m]1 and that of the regular polyball Bn(H), and differences regarding the
theory of free holomorphic (resp. pluriharmonic) functions on these noncommutative domains.
If A,B ∈ Bn(H)−, we say that A and B are Harnack equivalent (and denote A∼H B) if there exists
c>1 such that
1
F(rB)≤F(rA)≤c2F(rB), r ∈[0,1),
c2
for any positive free k-pluriharmonic function F : Bn(H) → B(E)⊗min B(H), where E is a separable
Hilbert space, in the sense of [29]. In this case, we write A∼HB. The equivalence classes with respect to
c
the equivalence relation ∼H are called Harnack parts of Bn(H)−. In Section 1, we prove a Harnack type
inequality for positive free k-pluriharmonic functions on regular polyballs and use it to show that the
Harnack part containing the zero element coincides with the open polyball Bn(H).
Given a Harnack part ∆ of Bn(H)−, we define the map δH :∆×∆→R+ by setting
δ (A,B):=lninf c>1: A∼HB .
H
c
n o
InSection2,weprovethatδ isametricon∆andprovideaSchwarz-Picktyperesultforfreeholomorphic
H
functionsonregularpolybalswithrespecttoδH. WeshowthatifΦ=(Φ1,...,Φm):Bn(H)→[B(H)m]−1
is a free holomorphic function on the regular polyball and X,Y ∈Bn(H), then Φ(X)∼HΦ(Y) and
δ (Φ(X),Φ(Y))≤δ (X,Y),
H H
whereδH isthehyperbolicmetricdefinedontheHarnackpartsof[B(H)m]−1 andonthepolyballBn(H),
respectively. Using the description of the group Aut(Bn) of all free holomorphic automorphisms of Bn
(see [28]), we prove that
δH(A,B)=δH(Ψ(A),Ψ(B)), Ψ∈Aut(Bn),
foranyA,B∈Bn(H)− suchthatA∼H B. Inparticular,thehyperbolicdistanceδH ontheopenpolyball
is invariant under the automorphism group Aut(Bn).
If A and B are in Bn(H)−, we say that they are Poisson equivalent (and denote A ∼P B) if there
exists c>1 such that
1
P(R,rB)≤P(R,rA)≤c2P(R,rB), r ∈[0,1),
c2
where X7→P(R,X) is the free pluriharmonic Poisson kernel on the regular polyball (see Section 1). In
thiscasewewriteA∼P B. Theequivalenceclasseswithrespecttoequivalencerelation∼H arecalledPoisson
c
partsofBn(H)−. WeproveinSection1thatthePoissonpartcontainingthe zeroelementcoincideswith
the open polyball Bn(H).
Given a Poissonpart ∆ of Bn(H)−, we define the map δP :∆×∆→R+ by setting
δ (A,B):=lninf c>1: A∼P B .
P
c
n o
InSection3,weprovethatδ isametricon∆andobtainanexplicitformulaforit(see Theorem3.3)in
P
termsofcertainnoncommutativeCauchykernelsactingontensorproductsoffullFockspaces. Moreover,
we prove that δP is a complete metric on Bn(H) and that the δP-topology coincides with the operator
norm topology on Bn(H).
In Section 4, we consider the regular polydisk Dk(H) := B (H), which consists of all tuples
(1,...,1)
X = (X1,...,Xk) of commuting strict contractions such that ∆X(I) > 0. We remark that in this case
we have
∆X(I)= (−1)p1+···+pkT1p1···Tkpk(Tk∗)pk···(T1∗)p1
p1,...,Xpk∈{0,1}
andDk(C)=Dk. Usingacharacterizationofpositivefreek-pluriharmonicfunctionsonregularpolydisks
(see Theorem 4.2) and the results of the previous sections, we prove that A∼HB if and only if A∼P B,
c c
4 GELUPOPESCU
for any A,B ∈ Dk(H)−. Consequently, the metrics δ and δ coincide on the Harnack (resp. Poisson)
H P
parts of Dk(H)−. We show that the hyperbolic metric δ is complete on the regular polydisk and the
H
δ -topology coincides with the operator norm topology on Dk(H). As a consequence, (Dk(H),δ ) is a
H H
complete hyperbolic space. Moreover, δ is invariant under the automorphism group Aut(Dk) and
H
δ (f(X),f(Y))≤δ (X,Y), X,Y ∈Dk(H)
H H
foranyfreeholomorphicfunctionf :Dk(H)→[B(H)m] . Therefore,thehyperbolicmetricδ onDk(H)
1 H
has similar properties to those of the Poincar´e distance on the open unit disc D. In addition, we prove
that if A and B are in Dk(H), then
δH(A,B)=lnmax CA(R)CB(R)−1 , CB(R)CA(R)−1 ,
where (cid:8)(cid:13) (cid:13) (cid:13) (cid:13)(cid:9)
k(cid:13) (cid:13) (cid:13) (cid:13)
CX(R):=(I ⊗∆X(I)1/2) (I−Ri⊗Xi∗), X=(X1,...,Xk)∈Dk(H),
i=1
Y
andR1,...Rk are the shift operatorson the Hardy space H2(Dk). In particular,we show that δH|Dk×Dk
is equivalent to the Kobayashidistance on the polydisk Dk (see [9]) and
1 k (1+|ψ (w )|)
δ (z,w)= ln i=1 zi i
H 2 k (1−|ψ (w )|)
Qi=1 zi i
for any z = (z1,...,zk) and w = (w1,...,wk)Qin Dk, where ψz := (ψz1,...,ψzn) is the involutive
automorphisms of Dk such that ψ (0)=z and ψ (z )=0.
zi i zi i
We remarkthat, accordingto the results ofthe presentpaper andthose from[21], the metrics δ and
H
δP coincide for the regular polydisk Dk(H) and for the noncommutative ball [B(H)n1]1. It remains an
open problem whether the same result is true for any regular polyball. On the other hand, it will be
interesting to see to what extent these results extend to more general polydomains in B(H)m such as
those studied in [24] and [26].
1. Harnack and Poisson equivalences on the closed polyball
In this section, we recall some basic facts concerning the noncommutative Berezin transforms on
polyballsandintroducetheHarnackandthePoissonequivalencerelationsontheclosedpolyballBn(H)−.
We providea Harnacktype inequality for positivefree k-pluriharmonicfunctions anduse itto show that
theHarnack(resp.Poisson)equivalenceclasscontainingthezeroelementcoincideswiththeopenpolyball
Bn(H).
Let H be an n -dimensional complex Hilbert space with orthonormalbasis ei,...,ei . We consider
ni i 1 ni
the full Fock space of H defined by F2(H ) := C1⊕ H⊗p, where H⊗p is the (Hilbert) tensor
ni ni p≥1 ni ni
product of p copies of H . Let F+ be the unital free semigroup on n generators gi,...,gi and the
ni ni L i 1 ni
identity gi. Set ei := ei ⊗···⊗ei if α = gi ···gi ∈ F+ and ei := 1 ∈ C. The length of α ∈ F+ is
0 α j1 jp j1 jp ni g0i ni
defined by |α| := 0 if α= gi and |α| :=p if α=gi ···gi , where j ,...,j ∈{1,...,n }. We define the
0 j1 jp 1 p i
left creation operator S acting on the Fock space F2(H ) by setting S ei := ei ⊗ei, α ∈ F+, and
i,j ni i,j α j α ni
the operator S acting on the Hilbert tensor product F2(H )⊗···⊗F2(H ) by setting
i,j n1 nk
S :=I⊗···⊗I⊗S ⊗I⊗···⊗I,
i,j i,j
i−1times k−itimes
where i ∈ {1,...,k} and j ∈ {1,...,ni}|. We{zdeno}te S := (|S1,.{.z.,Sk}), where Si := (Si,1,...,Si,ni), or
S:={Si,j}. ThenoncommutativeHardyalgebraF∞n (resp.thepolyballalgebraAn)istheweaklyclosed
(resp. normclosed) non-selfadjoint algebragenerated by {S } and the identity. Similarly, we define the
i,j
right creation operator R : F2(H ) → F2(H ) by setting R ei := ei ⊗ei for α ∈ F+, and the
i,j ni ni i,j α α j ni
operator R acting on the Hilbert tensor product F2(H )⊗···⊗F2(H ) by setting
i,j n1 nk
R :=I ⊗···⊗I⊗R ⊗I ⊗···⊗I.
i,j i,j
i−1times k−itimes
| {z } | {z }
HYPERBOLIC GEOMETRY ON NONCOMMUTATIVE POLYBALLS 5
ThepolyballalgebraRn isthenormclosednon-selfadjointalgebrageneratedby{Ri,j}andthe identity.
We recall (see [17], [26]) some basic properties for the noncommutative Berezin transforms associated
with regular polyballs. Let X = (X1,...,Xk) ∈ Bn(H)− with Xi := (Xi,1,...,Xi,ni). We use the
notation X := X ···X if α = gi ···gi ∈ F+ and X := I. The noncommutative Berezin
i,αi i,j1 i,jp i j1 jp ni i,g0i
kernel associated with any element X in the noncommutative polyball Bn(H)− is the operator
KX :H→F2(Hn1)⊗···⊗F2(Hnk)⊗∆X(I)(H)
defined by
KXh:= e1β1 ⊗···⊗ekβk ⊗∆X(I)1/2X1∗,β1···Xk∗,βkh, h∈H,
βi∈F+nXi,i=1,...,k
where ∆X(I) is the defect operator. A very important property of the Berezin kernel is that KXXi∗,j =
(S∗i,j⊗I)KX for anyi∈{1,...,k}andj ∈{1,...,ni}. The Berezin transform at X∈Bn(H)is the map
BX :B(⊗ki=1F2(Hni))→B(H) defined by
BX[g]:=K∗X(g⊗IH)KX, g ∈B(⊗ki=1F2(Hni)).
If g is in the C∗-algebra C∗(S) generated by S ,...,S , where i ∈ {1,...,k}, we define the Berezin
i,1 i,ni
transform at X ∈Bn(H)− by
BX[g]:= limK∗rX(g⊗IH)KrX, g ∈C∗(S),
r→1
where the limit is in the operator norm topology. In this case, the Berezin transform at X is a unital
completely positive linear map such that
BX(SαS∗β)=XαX∗β, α,β ∈F+n1 ×···×F+nk,
where S :=S ···S if α:=(α ,...,α )∈F+ ×···×F+ .
α 1,α1 k,αk 1 k n1 nk
The Berezin transforms will play an important role in this paper. More properties concerning non-
commutative Berezin transforms and multivariable operator theory on noncommutative balls and poly-
domains, can be found in [17] and [26]. For basic results on completely positive (resp. bounded) maps
we refer the reader to [15] and [16] .
Foreachm∈Z,wesetm+ :=max{m,0}andm− :=max{−m,0}. AfunctionF withoperator-valued
coefficients in B(E), where E is separable Hilbert space, is called free k-pluriharmonic on the abstract
polyball Bn if it has the form
F(X)= ··· A ⊗X ···X X∗ ···X∗ ,
(α1,...,αk;β1,...,βk) 1,α1 k,αk 1,β1 k,βk
mX1∈Z mXk∈Zαi,βi∈F+nXi,i∈{1,...,k}
|αi|=m−i ,|βi|=m+i
where the multi-series converge in the operator norm topology for any X = (X1,...,Xk) ∈ Bn(H),
with X := (X ,...,X ), and any Hilbert space H. Without loss of generality, we can assume
i i,1 i,ni
throughout this paper that H is a separable infinite dimensional Hilbert space. According to [29], the
order of the series in the definition above is irrelevant. Note that any free holomorphic function on Bn
is k-pluriharmonic. Indeed, according to [28], any free holomorphic function on the polyball Bn with
coefficients in B(E) has the form
f(X)= ··· C(α1,...,αk)⊗X1,α1···Xk,αk, X∈Bn(H),
mX1∈N mXk∈Nαi∈F+niX,i∈{1,...,k}
|αi|=mi
where the multi-series converge in the operator norm topology.
H
Now, we introduce a preorder relation ≺ on the closed ball Bn(H)−. If A and B are in Bn(H)−, we
H
say that A is Harnack dominated by B, and denote A≺ B, if there exists c>0 such that
F(rA)≤c2F(rB)
6 GELUPOPESCU
for any positive free k-pluriharmonic function F with operator valued coefficients and any r ∈ [0,1).
H H
When we want to emphasize the constant c, we write A≺B. Since ≺ is a preorder relation on Bn(H)−,
c
it induces an equivalence relation ∼H on Bn(H)−, which we call Harnack equivalence. The equivalence
classeswithrespectto∼H arecalledHarnack parts ofBn(H)−. LetAandBareinBn(H)−. Itis easyto
see that A and B are Harnack equivalent (we denote A∼H B) if and only if there exists c≥1 such that
1
(1.1) F(rB)≤F(rA)≤c2F(rB)
c2
for any positive free k-pluriharmonic function F with operator-valuedcoefficients and any r∈[0,1). We
also use the notation A∼HB if A≺HB and B≺HA.
c c c
We denote by C∗(S) the C∗-algebra generated by S , where i ∈ {1,...,k} and j ∈ {1,...,n }. A
i,j i
completely positive (c.p.) linear map µX : C∗(S) → B(H) is called representing c.p. map for the point
X∈Bn(H)− if
µX(S1,α1···Sk,αkS∗1,β1···S∗k,βk)=X1,α1···Xk,αkX∗1,β1···X∗k,βk
for any (α ,...,α ) and (β ,...,β ) in F+ ×···×F+ with α ,β ∈ F+, |α | = m−,|β | = m+, and
m ∈Z. 1 k 1 k n1 nk i i ni i i i i
i
Next, weobtaincharacterizationsforthe Harnackequivalence onthe closedregularpolyball Bn(H)−.
Proposition 1.1. Let A and B are in Bn(H)− and let c > 1. Then the following statements are
equivalent:
H
(i) A∼B;
c
(ii) c2BB−BA andc2BA−BBarecompletelypositivelinearmapontheoperatorspacespan{A∗nAn}−k·k,
where BX is the noncommutative Berezin transform at X∈Bn(H)−;
(iii) there are representing c.p. maps µA and µB for A and B, respectively, such that
1
µB ≤µA ≤c2µB.
c2
Proof. LetAandBbeelementsintheregularclosedpolyballBn(H)− andletc>1. Firstweprovethat
A≺HBifandonlyifc2BB−BAisacompletelypositivelinearmapontheoperatorspacespan{A∗nAn}−k·k.
c
Assume that A≺HB and let g ∈ P := span{A∗nAn}−k·k be a positive operator. According to Theorem
c
2.4 from [29], the map
F(X)=BX[g]:=K∗X[g⊗IH]KX, X∈Bn(H),
is a positive free k-pluriharmonic function on Bn(H) which has a continuous extension (in the operator
H
norm topology) to the closed ball Bn(H)−. Since A≺B, we have F(rA) ≤ c2F(rB) for any r ∈ [0,1),
c
which is equivalent to
c2BrB−BrA [g]≥0, r ∈[0,1).
Since BB[g]:=limr→1BrB[g]=F(cid:0)(B) exists in t(cid:1)he operator norm topology, and a similar result holds if
wereplaceBwithA,wededucethatc2BB−BA isapositivelinearmaponspan{A∗nAn}−k·k. Similarly,
passingtomatricesonecaprovethatc2BB−BAiscompletelypositive. Hence,wededucethatc2BB−BA
is completely positive on the operator space span{A∗nAn}−k·k.
Conversely, assume that c2BB−BA is completely positive on the operator space span{A∗nAn}−k·k.
Then c2BeBxt−BeAxt is positive on PE :=B(E)⊗minspan{A∗nAn}−k·k, where
BeXxt[g]:=(IE ⊗K∗X)[g⊗IH](IE ⊗KX), X∈Bn(H).
HYPERBOLIC GEOMETRY ON NONCOMMUTATIVE POLYBALLS 7
LetF :Bn(H)→B(E)⊗minB(H)beapositivefreek-pluriharmonicfunctiononBn(H). Then,foreach
r∈[0,1), we have
F(rX)=BeXxt[F(rS)] ≥0
and F(rS)∈P . Consequently, we have
E
c2F(rB)−F(rA)= c2BeBxt−BeAxt [F(rS)]≥0
for any r ∈[0,1), which proves our assertion. No(cid:0)w, the equivale(cid:1)nce of (i) and (ii) is clear.
It remains to prove that (ii) is equivalent to (iii). To this end, assume that item (ii) holds. According
to Arveson’s extension theorem [2], there are completely positive maps ϕ,ψ :C∗(S)→B(H) such that
ϕ(g)=c2BA[g]−BB[g] and ψ(g)=c2BB[g]−BA[g]
for any g ∈span{A∗nAn}−k·k. Hence, we deduce that
c2 c2
BA[g]= (c2ϕ(g)+ψ(g)) and BB[g]= (c2ψ(g)+ϕ(g)).
c4−1 c4−1
Now, we define µA :C∗(S)→B(H) and µB :C∗(S)→B(H) by setting
c2 c2
(1.2) µA := (c2ϕ+ψ) and µB := (c2ψ+ϕ)
c4−1 c4−1
and note that BA[g] = µA(g) and BB[g] = µB(g) for any g ∈ span{A∗nAn}−k·k. Due to the properties
of the noncommutative Berezin transform, it is clear that µA and µB are representing c.p. maps for A
and B, respectively. The inequalities 1µB ≤ µA ≤ c2µB are simple consequences of relation (1.2) and
c2
the fact that c > 1. To complete the proof, it is enough to prove that (iii) =⇒(i). To this end, assume
that (iii) holds for some c>1 and let F :Bn(H)→B(E)⊗minB(H) be a positive free k-pluriharmonic
function on Bn(H). Then F(rS)≥0 for any r ∈[0,1). Since F(rS)∈B(E)⊗minspan{A∗nAn}−k·k, we
deduce that
1
(id⊗µB)[F(rS)] ≤(id⊗µA)[F(rS)] ≤c2(id⊗µB)[F(rS)]
c2
for any r ∈[0,1), which implies A∼HB and completes the proof. (cid:3)
c
A bounded linear operator A ∈ B(E ⊗ k F2(H )) is called k-multi-Toeplitz with respect to the
i=1 ni
universal model R:=(R ,...,R ), where R :=(R ,...,R ), if
1 k Ni i,1 i,ni
(I ⊗R∗ )A(I ⊗R )=δ A, s,t∈{1,...,n },
E i,s E i,t st i
for every i ∈ {1,...,k}. Let Tn be the set of all k-multi-Toeplitz operators on E ⊗ ki=1F2(Hni). In
[29], we proved that
N
Tn =span{f∗g : f,g ∈B(E)⊗minAn}−SOT
=span{f∗g : f,g ∈B(E)⊗minAn}−WOT,
whereAn is the polyballalgebra. Inwhat follows,we providea Harnacktype inequality for positivefree
k-pluriharmonic function on the regular polyballs.
Theorem1.2. LetF beapositivefreek-pluriharmonicfunctionontheregularpolyball Bn,withoperator
coefficients in B(E) and let 0≤r <1. Then
1−r k 1+r k
F(0) ≤F(X)≤F(0)
1+r 1−r
(cid:18) (cid:19) (cid:18) (cid:19)
for any X∈rBn(H)−.
8 GELUPOPESCU
Proof. If F is a positive free k-pluriharmonic function on the regular polyball Bn, with operator coeffi-
cients in B(E), then there exist coefficients A ∈B(E) with α ,β ∈F+, |α |=m−,|β |=
(α1,...,αk;β1,...,βk) i i ni i i i
m+ such that, for any r ∈[0,1),
i
F(rS)= ··· A(α1,...,αk;β1,...,βk)⊗rPki=1(|αi|+|βi|)S1,α1S1∗,β1 ⊗···⊗Sk,αkSk∗,βk,
mX1∈Z mXk∈Zαi,βi∈F+nXi,i∈{1,...,k}
|αi|=m−i ,|βi|=m+i
where the multi-series is convergentin the operator norm topology and the sum does not depend on the
order of the series. In this case, F(rS) is a positive k-multi-Toeplitz operator and
F(rS)=F(rS ,...,rS )= C(α ;β )⊗r|αk|+|βk|S S∗ .
1 k k k k,αk k,βk
mXk∈Z|α|=m−kX,|βk|=m+k
Note that F(rS) is also a positive 1-multi-Topeplitz operator with respect to R = (R ,...,R ),
k k,1 k,nk
with coefficients in B(E)⊗ B(F2(H ))⊗ ···⊗ B(F2(H )). Applying the noncommutative
min n1 min min nk−1
Berezin transform at X =(X ,...,X ), we deduce that
k k,1 k,nk
G(X ):= C(α ;β )⊗r|αk|+|βk|X X∗ , X ∈[B(H)nk] ,
k k k k,αk k,βk k 1
mXk∈Z|α|=m−kX,|βk|=m+k
is a positive pluriharmonic function on the unit ball [B(H)nk]1. Using the Harnack type inequality from
[21], we deduce that
1−r 1+r
G(0) ≤G(rS )≤G(0) .
1+r k 1−r
NotethatG(rS )=F(rS ,...,rS )andG(0)=F(rS ,...,rS ,0)isapositive(k−1)-multi-Toeplitz
k 1 k 1 k−1
operator. As above, we obtain
1−r 1+r
F(rS ,...,rS ,0,0) ≤F(rS ,...,rS ,0)≤F(rS ,...,rS ,0,0) .
1 k−2 1+r 1 k−1 1 k−2 1−r
Continuing this process, we obtain
1−r 1+r
F(0,...,0) ≤F(rS ,0,...,0)≤F(0,...,0) .
1+r 1 1−r
Now, combining all these inequalities, we deduce that
1−r k 1+r k
F(0) ≤F(rS ,...,rS )≤F(0)
1+r 1 k 1−r
(cid:18) (cid:19) (cid:18) (cid:19)
foranyr ∈[0,1). ApplyingtheBerezintransformatX∈rBn(H)− tothelatterinequalities,wecomplete
the proof. (cid:3)
We define the free pluriharmonic Poisson kernel on the regular polyball Bn by setting
P(R,X):= ··· R∗ ···R∗ R ···R ⊗X ···X X∗ ···X∗
mX1∈Z mXk∈Zαi,βi∈F+nXi,i∈{1,...,k} 1,αe1 k,αek 1,βe1 k,βek 1,α1 k,αk 1,β1 k,βk
|αi|=m−i ,|βi|=m+i
for any X ∈ Bn(H), where the convergence is in the operator norm topology, and αi = giik···gii1 if
α = gi ···gi ∈ F+. According to [29], the map X 7→ P(R,X) is a positive free k-pluriharmonic
i i1 ik ni
function on Bn(H). In this case, Theorem 1.2 implies e
1−r k 1+r k
I ≤P(R,X)≤ I
1+r 1−r
(cid:18) (cid:19) (cid:18) (cid:19)
for any X∈rBn(H)−.
P
Now, we introduce a preorder relation ≺ on the closed ball Bn(H)−. If A and B are in Bn(H)−, we
P
say that A is Poisson dominated by B, and denote A≺ B, if there exists c>0 such that
P(R,rA)≤c2P(R,rB)
HYPERBOLIC GEOMETRY ON NONCOMMUTATIVE POLYBALLS 9
P P
for any r ∈ [0,1). When we want to emphasize the constant c, we write A≺B. Since ≺ is a preorder
c
relationonBn(H)−,itinduces anequivalencerelation∼P onBn(H)−,whichwe callPoisson equivalence.
The equivalence classes with respect to ∼P are called Poisson parts of Bn(H)−. Let A and B are in
Bn(H)−. It is easy to see that A and B are Poisson equivalent (we denote A∼P B) if and only if there
exists c≥1 such that
1
P(R,rB)≤P(R,rA)≤c2P(R,rB)
c2
for any r ∈[0,1). We also use the notation A∼P B if A≺P B and B≺P A. We remark that in the particular
c c c
case when k =1, the Poissonequivalence coincides with the Harnack equivalence (see [21]).
We recall that the spectral radius of an n -tuple A :=(A ,...,A ) of operators is given by
i i i,1 i,ni
r(A ):= lim k A A∗ k1/2p.
i p→∞ i,βi i,βi
βi∈F+nXi,|βi|=p
When n =1 we find again the usual spectral radius of an operator.
i
P
Lemma 1.3. Let A = (A1,...,Ak) ∈ Bn(H)−. Then A ≺ 0 if and only if the joint spectral radius
r(A )<1 for any i∈{1,...,k}.
i
P
Proof. Assume that A ≺ 0. Then there is c > 0 such that P(R,rA) ≤ c2I for any r ∈ [0,1). Set
w:= ei ⊗h ∈F2(H )⊗H, where h ∈H and kh k2 <∞, and let
α∈F+ni α α ni α α∈F+ni α
P w˜ := (1⊗···⊗1⊗ei ⊗P1⊗···⊗1)⊗h
α α
αX∈F+ni i−1times
be in F2(H )⊗···⊗F2(H ). Note th|at {z }
n1 nk
hP(R ,rA )w,wi=hP(R,rA)w˜,w˜i≤c2kw˜k2 =c2kwk2
i i
foranyw∈F2(H )⊗H,whereP(R ,rA )is the Poissonkernelassociatewiththe rowcontractionrA .
ni i i i
P
Applying Theorem 1.2 from [21], we deduce that A ≺ 0 and, consequently, r(A )<1.
i i
Conversely, let A = (A1,...,Ak) ∈ Bn(H)− and assume that r(Ai) < 1 for any i ∈ {1,...,k}. Due
to the fact that, for each i∈{1,...,k}, R ,...,R are isometries with orthogonalranges, we have
i,1 i,ni
r(A )=r(R ⊗A∗ +···+R ⊗A∗ ).
i i,1 i,1 i,ni i,ni
Setting Λ :=R∗ ⊗A +···+R∗ ⊗A we have r(Λ )=r(A )<1 and deduce that the spectrum
i i,1 i,1 i,ni i,ni i i
of Λ is included in D:={z ∈C: |z|<1}. Since λ7→(λI −Λ )−1 is continuous on the resolvent ρ(Λ ),
i i i
one can easily see that M :=sup k(I−rΛ )−1k<∞. According to [29], P(R,rS) is equal to
i r∈[0,1) i
k k
(I −R∗i,1⊗rSi,1−···−R∗i,ni ⊗rSi,ni)−1 (I ⊗∆rS(I)) (I−Ri,1⊗rS∗i,1−···−Ri,ni⊗rS∗i,ni)−1
" #
i=1 i=1
Y Y
for any r ∈ [0,1). Since the noncommutative Berezin transform BA is continuous in the operator norm
and completely positive, so is id⊗BA. Using the map id⊗BA, we deduce that
P(R,rA)=(id⊗BA)[P(R,rS)]
k k
= (I−rΛi)−1(I⊗∆rA(I)) (I −rΛ∗i)−1
i=1 i=1
Y Y
≤M2···M2I
1 k
for any r ∈[0,1). This shows that A≺P 0 and completes the proof. (cid:3)
10 GELUPOPESCU
Now,weshowthattheHarnack(resp.Poisson)equivalenceclasscontainingthezeroelementcoincides
with the open polyball Bn(H).
Theorem 1.4. Let A=(A1,...,Ak)∈Bn(H)−. Then the following statements are equivalent.
(i) A∼H 0;
(ii) r(A )<1 for any i∈{1,...,k} and there exists a>0 such that
i
P(R,rA)≥aI, r ∈[0,1);
(iii) A∈Bn(H);
(iv) A∼P 0.
Proof. Note that if F : Bn(H) → B(E)⊗min B(H) is a free k-pluriharmonic function then, for each
i∈{1,...,k}, the map
X 7→F(0,...,0,X ,0,...,0), X ∈[B(H)ni] ,
i i i 1
i−1times
is a pluriharmonic function on [B(H| ){nzi]1}. Conversely, if Fi : [B(H)ni]1 → B(E)⊗min B(H) is a free
pluriharmonic function on the open unit ball [B(H)ni]1, then the map X= (X1,...,Xk)7→ Fi(Xi) is a
free k-pluriharmonic function on Bn(H). Consequently, if A = (A1,...,Ak) and B = (B1,...,Bk) are
in the closed regular polyball Bn(H)− and A∼H B, then Ai ∼H Bi for any i∈{1,...,k}.
Now, assume that A ∼H 0. Due to the remark above, we must have A ∼H 0 for any i ∈ {1,...,k}.
i
Applying Theorem 1.6 from [21] to Ai, we deduce that Ai ∈[B(H)ni]1. In particular, the joint spectral
radius r(A ) < 1. Since A ∼H 0 and the map X 7→ P(R,X) is a positive free k-pluriharmonic function
i
on Bn(H), there is a>0 such that P(R,rA)≥aI for any r ∈[0,1). Therefore, item (ii) holds.
Now, we prove that (ii)=⇒(iii). As in the proof of Lemma 1.3, we have
k k
P(R,rA)= (I −rΛi)−1(I ⊗∆rA(I)) (I−rΛ∗i)−1,
i=1 i=1
Y Y
where Λ :=R∗ ⊗A +···+R∗ ⊗A . Moreover,the spectral radius of the operator Λ coincides
i i,1 i,1 i,ni i,ni i
with r(A )<1 and
i
k
0<M := sup k (I−rΛ )−1k<∞.
i
r∈[0,1) i=1
Y
Hence and using the fact that P(R,rA)≥aI for any r ∈[0,1), we obtain
c
I ⊗∆rA(I)≥ M2I ⊗I, r ∈[0,1).
Hence∆A(I)≥ c I,whichshowsthatA∈Bn(H). Therefore,item(iii)holds. Toprovetheimplication
M2
(iii)=⇒(i), assume that A∈Bn(H). Define
γA =mBn(A):=inf{t>0: A∈tBn(H)}.
Due to Proposition 2.6 from [29], we have mBn(A) < 1 and, consequently, rA ∈ γABn(H)− for any
r∈[0,1). Applying the Harnack type inequality of Theorem 1.2, we obtain the inequalities
1−γA k 1+γA k
F(0) ≤F(rA)≤F(0)
1+γA 1−γA
(cid:18) (cid:19) (cid:18) (cid:19)
for any positive free k-pluriharmonic function F on Bn(H) and any r ∈ [0,1). Consequently, A ∼H 0,
which completes the proof of the implication (iii) =⇒ (i). Note that if A ∼H 0, then we automatically
P
have A ∼ 0. On the other hand, taking into account Lemma 1.3, one can easily see that (iv) =⇒ (ii).
This completes the proof. (cid:3)