Characteristic Functions for Ergodic Tuples Santanu Dey and Rolf Gohm Abstract. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a definition given by G.Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed. Mathematics Subject Classification (2000). Primary 47A20, 47A13; Secondary 46L53, 46L05. Keywords. completely positive, dilation, conjugacy, ergodic, coisometric, row contraction, characteristic function, Cuntz algebra. 0. Introduction If Z = summationtextdi=1 Ai ? A?i is a normal, unital, ergodic, completely positive map on B(H), the bounded linear operators on a complex separable Hilbert space, and if there is a (necessarily unique) invariant vector state for Z, then we also say that A = (A1,...,Ad) is a coisometric, ergodic row contraction with a one-dimensional invariant subspace for the adjoints. Precise definitions are given below. This is the main setting to be investigated in this paper. In Section 1 we give a concise review of a result on the dilations of Z obtained by R.Gohm in [Go04] in a chapter called ?Cocycles and Coboundaries?. There exists a conjugacy between a homomorphic dilation of Z and a tensor shift, and we emphasize an explicit infinite product formula that can be obtained for the intertwining unitary. [Go04] may also be consulted for connections of this topic to a scattering theory for noncommutative Markov chains by B.K?ummerer and H.Maassen (cf. [KM00]) and more general for the relevance of this setting in applications. In this work we are concerned with its relevance in operator theory and correspondingly in Section 2 we shift our attention to the row contraction A = (A1,...,Ad). Our starting point has been the observation that the intertwining unitary mentioned above has many similarities with the notion of characteristic 2 Santanu Dey and Rolf Gohm function occurring in the theory of functional models of contractions, as initiated by B.Sz.-Nagy and C.Foias (cf. [NF70, FF90]). In fact, the center of our work is the commuting diagram 3.3 in Section 3, which connects the results in [Go04] mentioned above with the theory of minimal isometric dilations of row contractions by G.Popescu (cf. [Po89a]) and shows that the intertwining unitary determines a multi-analytic inner function, in the sense introduced by G.Popescu in [Po89c, Po95]. We call this inner function the extended characteristic function of the tuple A, see Definition 3.3. Section 4 is concerned with an explicit computation of this inner function. In Section 5 we show that it is an extension of the characteristic function of the ?-stable part ?A of A, the latter in the sense of Popescu?s generalization of the Sz.-Nagy-Foias theory to row contractions (cf. [Po89b]). This explains why we call our inner function an extended characteristic function. The row contraction A is a one-dimensional extension of the ?-stable row contraction ?A, and in our analysis we separate the new part of the characteristic function from the part already given by Popescu. G.Popescu has shown in [Po89b] that for completely non-coisometric tuples, in particular for ?-stable ones, his characteristic function is a complete invariant for unitary equivalence. In Section 6 we prove that our extended characteristic function does the same for the tuples A described above. In this sense it is char- acteristic. This is remarkable because the strength of Popescu?s definition lies in the completely non-coisometric situation while we always deal with a coisometric tuple A. The extended characteristic function also does not depend on the choice of the decompositionsummationtextdi=1 Ai?A?i of the completely positive map Z and hence also characterizes Z up to conjugacy. We think that together with its nice properties established earlier this clearly indicates that the extended characteristic function is a valuable tool for classifying and investigating such tuples respectively such completely positive maps. Section 7 contains a worked example for the constructions in this paper. 1. Weak Markov dilations and conjugacy In this section we give a brief and condensed review of results in [Go04], Chapter 2, which will be used in the following and which, as described in the introduction, motivated the investigations documented in this paper. We also introduce notation. A theory of weak Markov dilations has been developed in [BP94]. For a (sin- gle) normal unital completely positive map Z : B(H) ? B(H), where B(H) consists of the bounded linear operators on a (complex, separable) Hilbert space, it asks for a normal unital ??endomorphism ?J : B( ?H) ? B( ?H), where ?H is a Hilbert space containing H, such that for all n ?N and all x ? B(H) Zn(x) = pH ?Jn(xpH) |H. Characteristic Functions for Ergodic Tuples 3 Here pH is the orthogonal projection onto H. There are many ways to construct ?J. In [Go04], 2.3, we gave a construction analogous to the idea of ?coupling to a shift? used in [K?u85] for describing quantum Markov processes. This gives rise to a number of interesting problems which remain hidden in other constructions. We proceed in two steps. First note that there is a Kraus decomposition Z(x) = summationtextdi=1 ai xa?i with (ai)di=1 ? B(H). Here d = ? is allowed in which case the sum should be interpreted as a limit in the strong operator topology. Let P be a d-dimensional Hilbert space with orthonormal basis {epsilon11,...,epsilon1d}, further K another Hilbert space with a distinguished unit vector ?K ? K. We identify H with H??K ? H?K and again denote by pH the orthogonal projection onto H. For K large enough there exists an isometry u : H?P ? H?K s.t. pHu(h?epsilon1i) = ai(h), for all h ? H, i = 1,...,d, or equivalently, u?(h??K) = dsummationdisplay i=1 a?i(h)?epsilon1i. Explicitly, one may take K = Cd+1 (resp. infinite-dimensional) and identify H?K similarequal (H??K)? dcircleplusdisplay 1 H similarequal H? dcircleplusdisplay 1 H. Then, using isometries u1,...,ud : H ? H?circleplustextd1 H with orthogonal ranges and such that ai = pHui for all i (for example, such isometries are explicitly constructed in Popescu?s formula for isometric dilations, cf. [Po89a] or equation 3.2 in Section 3), we can define u(h?epsilon1i) := ui(h) for all h ? H, i = 1,...,d and check that u has the desired properties. Now we define a ??homomorphism J : B(H) ? B(H?K), x mapsto? u(x?1P)u?. It satisfies pHJ(x)(h??K) = pHu(x?1)u?(h??K) = pHu(x?1)parenleftbig dsummationdisplay i=1 a?i(h)?epsilon1iparenrightbig= dsummationdisplay i=1 ai xa?i(h) = Z(x)(h), which means that J is a kind of first order dilation for Z. 4 Santanu Dey and Rolf Gohm For the second step we write ?K := circlemultiplytext?1 K for an infinite tensor product of Hilbert spaces along the sequence (?K) of unit vectors in the copies of K. We have a distinguished unit vector ??K and a (kind of) tensor shift R : B(?K) ? B(P ? ?K), ?y mapsto?1P ? ?y. Finally ?H := H? ?K and we define a normal ??endomorphism ?J : B( ?H) ? B( ?H), B(H)?B(?K) owner x? ?y mapsto? J(x)? ?y ? B(H?K)?B(?K). Here we used von Neumann tensor products and (on the right hand side) a shift identification K? ?K similarequal ?K. We can also write ?J in the form ?J(?) = u(IdH ?R)(?)u?, where u is identified with u?1?K. The natural embedding H similarequal H???K ? ?H leads to the restriction ?J := ?J| ?H with ?H := spann?0 ?Jn(pH)( ?H), which can be checked to be a normal unital ?-endomorphism satisfying all the properties of a weak Markov dilation for Z described above. See [Go04], 2.3. A Kraus decomposition of ?J can be written as ?J(x) = dsummationdisplay i=1 ti xt?i, where ti ? B( ?H) is obtained by linear extension of H? ?K owner h? ?k mapsto? ui(h)? ?k = u(h?epsilon1i)??k ? (H?K)??K similarequal H??K. Because ?J is a normal unital ??endomorphism the (ti)di=1 generate a representation of the Cuntz algebra Od on ?H which we called a coupling representation in [Go04], 2.4. Note that the tuple (t1,...,td) is an iso- metric dilation of the tuple (a1,...,ad), i.e., the ti are isometries with orthogonal ranges and pHtni |H = ani for all i = 1,...,d and n ?N. The following multi-index notation will be used frequently in this work. Let ? denote the set {1,2,...,d}. For operator tuples (a1,...,ad), given ? = (?1,...,?m) in ?m, a? will stand for the operator a?1a?2 ...a?m, |?| := m. Further ?? := ??n=0?n, where ?0 := {0} and a0 is the identity operator. If we write a?? this always means (a?)? = a??m ...a??1. Back to our isometric dilation, it can be checked that span{t?h : h ? H,? ? ??} = ?H, which means that we have a minimal isometric dilation, cf. [Po89a] or the begin- ning of Section 3. For more details on the construction above see [Go04], 2.3 and 2.4. Assume now that there is an invariant vector state for Z : B(H) ? B(H) given by a unit vector ?H ? H. Equivalent: There is a unit vector ?P =summationtextdi=1 ?iepsilon1i ? P such that u(?H ? ?P) = ?H ? ?K. Also equivalent: For i = 1,...,d we have Characteristic Functions for Ergodic Tuples 5 a?i ?H = ?i ?H. Here ?i ?Cwithsummationtextdi=1|?i|2 = 1 and we used complex conjugation to get nice formulas later. See [Go04], A.5.1, for a proof of the equivalences. On ?P := circlemultiplytext?1 P along the unit vectors (?P) in the copies of P we have a tensor shift S : B( ?P) ? B( ?P), ?y mapsto?1P ? ?y. Its Kraus decomposition is S(?y) =summationtextdi=1 si ?ys?i with si ? B( ?P) and si(?k) = epsilon1i ??k for ?k ? ?P and i = 1,...,d. In [Go04], 2.5, we obtained an interesting description of the situation when the dilation ?J is conjugate to the shift endomorphism S. This result will be further analyzed in this paper. We give a version suitable for our present needs but the reader should have no problems to obtain a proof of the following from [Go04], 2.5. Theorem 1.1. Let Z : B(H) ? B(H) be a normal unital completely positive map with an invariant vector state ??H,??H?. Notation as introduced above, d ? 2. The following assertions are equivalent: (a) Z is ergodic, i.e., the fixed point space of Z consists of multiples of the iden- tity. (b) The vector state ??H,??H? is absorbing for Z, i.e., if n ? ? then ?(Zn(x)) ? ??H,x?H? for all normal states ? and all x ? B(H). (In particular, the in- variant vector state is unique.) (c) ?J and S are conjugate, i.e., there exists a unitary w : ?H ? ?P such that ?J(?x) = w?S(w?xw?)w. (d) The Od?representations corresponding to ?J and S are unitarily equivalent, i.e., wti = siw for i = 1,...,d. An explicit formula can be given for an intertwining unitary as occurring in (c) and (d). If any of the assertions above is valid then the following limit exists strongly, ?w = limn??u?0n ...u?01 : H? ?K ? H? ?P, where we used a leg notation, i.e., u0n = (IdH ?R)n?1(u). In other words u0n is u acting on H and on the n?th copy of P. Further ?w is a partial isometry with initial space ?H and final space ?P similarequal ?H ? ?P ? H? ?P and we can define w as the corresponding restriction of ?w. To illustrate the product formula for w, which will be our main interest in this work, we use it to derive (d). wti(h? ?k) = wbracketleftbigu(h?epsilon1i)? ?kbracketrightbig= limn??u?0n ...u?01u01(h?epsilon1i ? ?k) = limn??u?0n ...u?02(h?epsilon1i ? ?k) = siw(h? ?k). Let us finally note that Theorem 1.1 is related to the conjugacy results in [Pow88] and [BJP96]. Compare also Proposition 2.4. 6 Santanu Dey and Rolf Gohm 2. Ergodic coisometric row contractions In the previous section we considered a map Z : B(H) ? B(H) given by Z(x) =summationtext d i=1 Ai xA ?i, where (Ai)di=1 ? B(H). We can think of (Ai)di=1 as a d-tuple A = (A1,...,Ad) or (with the same notation) as a linear map A = (A1,...,Ad) : dcircleplusdisplay i=1 H ? H. (Concentrating now on the tuple we have changed to capital letters A. We will sometimes return to lower case letters a when we want to emphasize that we are in the (tensor product) setting of Section 1.) We have the following dictionary. Z(1) ?1 ? dsummationdisplay i=1 Ai A?i ?1 ? A is a contraction Z(1) = 1 ? dsummationdisplay i=1 Ai A?i = 1 parenleftbigZ is called unitalparenrightbig parenleftbigA is called coisometricparenrightbig ??H,??H? = ??H,Z(?)?H? ? A?i ?H = ?i ?H, ?i ?C, dsummationdisplay i=1 |?i|2 = 1 parenleftbig invariant vector stateparenrightbig parenleftbig common eigenvector for adjointsparenrightbig Z ergodic ? {Ai,A?i}prime = C1parenleftbig trivial fixed point spaceparenrightbig parenleftbigtrivial commutantparenrightbig The converse of the implication at the end of the dictionary is not valid. This is related to the fact that the fixed point space of a completely positive map is not always an algebra. Compare the detailed discussion of this phenomenon in [BJKW00]. By a slight abuse of language we call the tuple (or row contraction) A = (A1,...,Ad) ergodic if the corresponding map Z is ergodic. With this terminology we can interpret Theorem 1.1 as a result about ergodic coisometric row contrac- tions A with a common eigenvector ?H for the adjoints A?i. This will be examined starting with Section 3. To represent these objects more explicitly let us write ?H:= HcircleminusC? H. With respect to the decomposition H = C?H? ?H we get 2?2? block matrices Ai = parenleftbigg ? i 0 |lscripti? ?Ai parenrightbigg , A?i = parenleftbigg ? i ?lscripti| 0 ?A?i parenrightbigg . (2.1) Characteristic Functions for Ergodic Tuples 7 Here ?Ai ? B( ?H) and lscripti ??H. For the off-diagonal terms we used a Dirac notation that should be clear without further comments. Note that the case d = 1 is rather uninteresting in this setting because if A is a coisometry with block matrix parenleftbigg ? 0 |lscript? ?A parenrightbigg then because parenleftbigg 1 0 0 1 parenrightbigg = AA? = parenleftbigg |?|2 ??lscript| ?|lscript? |lscript??lscript|+ ?A?A? parenrightbigg we always have lscript = 0. But for d ? 2 there are many interesting examples arising from unital ergodic completely positive maps with invariant vector states. See Section 1 and also Section 7 for an explicit example. We always assume d ? 2. Proposition 2.1. A coisometric row contraction A = (A1,...,Ad) is ergodic with common eigenvector ?H for the adjoints A?1,...,A?d if and only if ?H is invariant for A1,...,Ad and the restricted row contraction (?A1,...,?Ad) on ?H is ?-stable, i.e., for all h ??H limn?? summationdisplay |?|=n bardbl?A??hbardbl2 = 0 . Here we used the multi-index notation introduced in Section 1. Note that ?-stable tuples are also called pure, we prefer the terminology from [FF90]. Proof. It is clear that ?H is a common eigenvector for the adjoints if and only if ?H is invariant for A1,...,Ad. Let Z(?) = summationtextdi=1 Ai ?A?i be the associated completely positive map. With q := 1?|?H???H|, the orthogonal projection onto ?H, and by using qAi q = Ai q similarequal ?Ai for all i, we get Zn(q) = summationdisplay |?|=n A? qA?? = summationdisplay |?|=n ?A? ?A?? and thus for all h ??H summationdisplay |?|=n bardbl?A??hbardbl2 = ?h,Zn(q)h?. Now it is well known that ergodicity of Z is equivalent to Zn(q) ? 0 for n ? ? in the weak operator topology. See [GKL06], Prop. 3.2. This completes the proof. a50 Remark 2.2. Given a coisometric row contraction a = (a1,...,ad) we also have the isometry u : H?P ? H?K from Section 1. We introduce the linear map a : P ? B(H), k mapsto? ak defined by a?k(h)?k := (1H ?|k??k|)u?(h??K). Compare [Go04], A.3.3. In particular ai = aepsilon1i for i = 1,...,d, where {epsilon11,...,epsilon1d} is the orthonormal basis of P used in the definition of u. Arveson?s metric operator 8 Santanu Dey and Rolf Gohm spaces, cf. [Ar03], give a conceptual foundation for basis transformations in the operator space linearly spanned by the ai. Similarly, in our formalism a unitary in B(P) transforms a = (a1,...,ad) into another tuple aprime = (aprime1,...,aprimed). If ?H is a common eigenvector for the adjoints a?i then ?H is also a common eigenvector for the adjoints (aprimei)? but of course the eigenvalues are transformed to another tuple ?prime = (?prime1,...,?primed). We should consider the tuples a and aprime to be essentially the same. This also means that the complex numbers ?i are not particularly important and they should not play a role in classification. They just reflect a certain choice of orthonormal basis in the relevant metric operator space. Independent of basis transformations is the vector ?P = summationtextdi=1 ?i epsilon1i ? P satisfying u(?H ? ?P) = ?H ??K (see Section 1) and the operator a?P =summationtextdi=1 ?i ai. For later use we show Proposition 2.3. Let A = (A1,...,Ad) be an ergodic coisometric row contraction such that A?i ?H = ?i ?H for all i, further A?P := summationtextdi=1 ?i Ai. Then for n ? ? in the strong operator topology (A??P)n ? |?H???H|. Proof. We use the setting of Section 1 to be able to apply Theorem 1.1. From u?(h??K) =summationtextdi=1 a?i(h)?epsilon1i we obtain u?(h??K) = a??P(h)??P ?hprime with hprime ? H ? ??P. Assume that h ??H. Because u? is isometric on H ? ?K we conclude that u?(?H ??K) = ?H ??P ? u?(h??K) (2.2) and thus also a??P(h) ??H. In other words, a??P( ?H) ? ?H . Let qn be the orthogonal projection from H ?circlemultiplytextn1 P onto ?H ?circlemultiplytextn1 P. From Theorem 1.1 it follows that (1?qn)u?0n ...u?01(h? ncirclemultiplydisplay 1 ?K) ? 0 (n ? ?). On the other hand, by iterating the formula from the beginning, u?0n ...u?01(h? ncirclemultiplydisplay 1 ?K) =parenleftbig(a??P)n(h)? ncirclemultiplydisplay 1 ?Pparenrightbig?hprime with hprime ? H?(circlemultiplytextn1 ?P)?. It follows that also (1?qn)parenleftbig(a??P)n(h)? ncirclemultiplydisplay 1 ?Pparenrightbig? 0. Characteristic Functions for Ergodic Tuples 9 But from a??P( ?H) ? ?H we have qnparenleftbig(a??P)n(h)?circlemultiplytextn1 ?Pparenrightbig= 0 for all n. We conclude that (a??P)n(h) ? 0 for n ? ?. Further a??P?H = dsummationdisplay i=1 ?i a?i ?H = dsummationdisplay i=1 ?i ?i ?H = ?H, and the proposition is proved. a50 The following proposition summarizes some well known properties of minimal isometric dilations and associated Cuntz algebra representations. Proposition 2.4. Suppose A is a coisometric tuple on H and V is its minimal isometric dilation. Assume ?H is a distinguished unit vector in H and ? = (?1,...,?d) ?Cd, summationtexti|?i|2 = 1. Then the following are equivalent. 1. A is ergodic and A?i ?H = ?i ?H for all i. 2. V is ergodic and V?i ?H = ?i ?H for all i. 3. V?i ?H = ?i ?H and V generates the GNS-representation of the Cuntz algebra Od = C?{g1,??? ,gd} (gi its abstract generators) with respect to the Cuntz state which maps g? g?? mapsto? ?? ??, ??,? ? ??. Cuntz states are pure and the corresponding GNS-representations are irreducible. This Proposition clearly follows from Theorem 5.1 of [BJKW00], Theorem 3.3 and Theorem 4.1 of [BJP96]. Note that in Theorem 1.1(d) we already saw a concrete version of the corresponding Cuntz algebra representation. 3. A new characteristic function First we recall some more details of the theory of minimal isometric dilations for row contractions (cf. [Po89a]) and introduce further notation. The full Fock space over Cd (d ? 2) denoted by ?(Cd) is ?(Cd) := C?Cd ?(Cd)?2 ?????(Cd)?m ???? . 1?0???? is called the vacuum vector. Let{e1,...,ed}be the standard orthonormal basis of Cd. Recall that we include d = ? in which case Cd stands for a complex separable Hilbert space of infinite dimension. For ? ? ??, e? will denote the vector e?1 ?e?2 ?????e?m in the full Fock space ?(Cd) and e0 will denote the vacuum vector. Then the (left) creation operators Li on ?(Cd) are defined by Lix = ei ?x for 1 ? i ? d and x ? ?(Cd). The row contraction L = (L1,...,Ld) consists of isometries with orthogonal ranges. 10 Santanu Dey and Rolf Gohm Let T = (T1,??? ,Td) be a row contraction on a Hilbert space H. Treating T as a row operator from circleplustextdi=1H to H, define D? := (1?TT?)12 : H ? H and D := (1?T?T)12 :circleplustextdi=1H ?circleplustextdi=1H. This implies that D? = (1? dsummationdisplay i=1 TiT?i )12, D = (?ij1?T?i Tj)12d?d. (3.1) Observe that TD2 = D2?T and hence TD = D?T. Let D := Range D and D? := Range D?. Popescu in [Po89a] gave the following explicit presentation of the minimal isometric dilation of T by V on H?(?(Cd)?D), Vi(h? summationdisplay ???? e? ?d?) = Tih?[e0 ?Dih+ei ? summationdisplay ???? e? ?d?] (3.2) for h ? H and d? ? D. Here Dih := D(0,...,0,h,0,...,0) and h is embedded at the ith component. In other words, the Vi are isometries with orthogonal ranges such that T?i = V?i |H for i = 1,...,d and the spaces V?H with ? ? ?? together span the Hilbert space on which the Vi are defined. It is an important fact, which we shall use repeatedly, that such minimal isometric dilations are unique up to unitary equiv- alence (cf. [Po89a]). Now, as in Section 2, let A = (A1,??? ,Ad), Ai ? B(H), be an ergodic coisometric tuple with A?i?H = ?i?H for some unit vector ?H ? H and some ? ? Cd, summationtexti|?i|2 = 1. Let V = (V1,??? ,Vd) be the minimal isometric dilation of A given by Popescu?s construction (see equation 3.2) onH?parenleftbig?(Cd)?DAparenrightbig. Because A?i = V?i |H we also have V?i ?H = ?i?H and because V generates an irreducible Od?representation (Proposition 2.4), we see that V is also a minimal isometric dilation of ? : Cd ?C. In fact, we can think of ? as the most elementary example of a tuple with all the properties stated for A. Let ?V = (?V1,??? , ?Vd) be the minimal isometric dilation of ? given by Popescu?s construction on C?(?(Cd)?D?). Because A is coisometric it follows from equation 3.1 that D is in fact a projection and hence D = (?ij1?A?iAj)d?d. We infer that D(A?1,??? ,A?d)T = 0, where T stands for transpose. Applied to ? instead of A this shows that D? = (1?|????|) and D? ?C(?1,??? ,?d)T = Cd, where ? = (?1,??? ,?d). Remark 3.1. Because ?H is cyclic for {V?, ? ? ??} we have span{A??H : ? ? ??} = span{pHV??H : ? ? ??} = H. Using the notation from equation 2.1 this further implies that span{?A? li : ? ? ??,1 ? i ? d} = ?H . Characteristic Functions for Ergodic Tuples 11 As minimal isometric dilations of the tuple ? are unique up to unitary equi- valence, there exists a unitary W : H?(?(Cd)?DA) ?C?(?(Cd)?D?), such that WVi = ?ViW for all i. After showing the existence of W we now proceed to compute W explicitly. For A, by using Popescu?s construction, we have its minimal isometric dilation V on H?(?(Cd)?DA). Another way of constructing a minimal isometric dilation t of a was demonstrated in Section 1 on the space ?H (obtained by restricting to the minimal subspace of H? ?K with respect to t). Identifying A and a on the Hilbert space H there is a unitary ?A : ?H ? H?(?(Cd)?DA) which is the identity on H and satisfies Vi?A = ?Ati. By Theorem 1.1(d) the tuple s on ?P arising from the tensor shift is unitarily equivalent to t (resp. V), explicitly wti = siw for all i. An alternative view- point on the existence of w is to note that s is a minimal isometric dilation of ?. In fact, s?i ??P = ?epsilon1i,?P???P = ?i ??P for all i. Hence there is also a unitary ?? : ?P ?C?(?(Cd)?D?) with ????P = 1 ?C which satisfies ?Vi?? = ??si. Remark 3.2. It is possible to describe ?? in an explicit way and in doing so to construct an interesting and natural (unitary) identification of circlemultiplytext?1 Cd and C?(?(Cd)?Cd?1). In fact, recall (from Section 1) that ?P =circlemultiplytext?1 P and the space P is nothing but a d-dimensional Hilbert space. Hence we can identify Cd similarequal P = ?P ?C?P similarequal D? ?C?T similarequalCd?1 ?C In this identification the orthonormal basis (epsilon1i)di=1 of P goes to the canonical basis (ei)di=1 of Cd, in particular the vector ?P = summationtexti ?i epsilon1i goes to ?T = (?1,??? ,?d)T and we have ?Psimilarequal D?. Then we can write ?? : ??P mapsto? 1 ?C, k ???P mapsto? e0 ?k epsilon1? ?k ???P mapsto? e? ?k, where k ??P, ? ? ??, epsilon1? = epsilon1?1?...epsilon1?n ?circlemultiplytextn1 P (the first n copies ofP in the infinite tensor product ?P), e? = e?1 ?...e?n ? ?(Cd) as usual. It is easily checked that ?? given in this way indeed satisfies the equation ?Vi?? = ??si (for all i), which may thus be seen as the abstract characterization of this unitary map (together with ????P = 1). Summarizing, for i = 1,...,d Vi ?A = ?A ti, wti = siw, ?Vi ?? = ?? si 12 Santanu Dey and Rolf Gohm and we have the commuting diagram ?H w d47d47 ?A d15d15 ?P ?? d15d15 H?(?(Cd)?DA) W d47d47 C?(?(Cd)?D?). (3.3) From the diagram we get W = ??w??1A . Combined with the equations above this yields WVi = ?Vi W and we see that W is nothing but the dilations-intertwining map which we have already introduced earlier. Hence w and W are essentially the same thing and for the study of certain problems it may be helpful to switch from one picture to the other. In the following we analyze W to arrive at an interpretation as a new kind of characteristic function. First we have an isometric embedding ?C := W|H : H ?C?(?(Cd)?D?). (3.4) Note that ?C ?H = W ?H = 1 ?C. The remaining part is an isometry M?? := W|?(Cd)?DA : ?(Cd)?DA ? ?(Cd)?D?. (3.5) From equation 3.2 we get for all i Vi|?(Cd)?DA = (Li ?1DA), ?Vi|?(Cd)?D ? = (Li ?1D?), and we conclude that M??(Li ?1DA) = (Li ?1D?)M??, ?1 ? i ? d. (3.6) In other words, M?? is a multi-analytic inner function in the sense of [Po89c, Po95]. It is determined by its symbol ?? := W|e 0?DA : DA ? ?(C d)?D?, (3.7) where we have identified e0?DA and DA. In other words, we think of the symbol ?? as an isometric embedding of DA into ?(Cd)?D?. Definition 3.3. We call M?? (or ??) the extended characteristic function of the row contraction A, See Sections 5 and 6 for more explanation and justification of this terminol- ogy. Characteristic Functions for Ergodic Tuples 13 4. Explicit computation of the extended characteristic function To express the extended characteristic function more explicitly in terms of the tuple A we start by defining ?D? : ?H= HcircleminusC?H ? ?P = P circleminusC?P similarequal D?, (4.1) h mapsto?parenleftbig??H|?1Pparenrightbigu?(h??K), where u : H?P ? H?K is the isometry introduced in Section 1. That indeed the range of ?D? is contained in ?P follows from equation 2.2, i.e., u?(h??K) ? ?H??P for h ??H. With notations from equation 2.1 we can get a more concrete formula. Lemma 4.1. For all h ??H we have ?D?(h) =summationtextdi=1?lscripti,h?epsilon1i. Proof. parenleftbig??H|?1Pparenrightbigu?(h??K) =summationtextdi=1??H,a?ih??epsilon1i =summationtextdi=1?lscripti,h?epsilon1i. a50 Proposition 4.2. The map ?C : H ? C?(?(Cd)?D?) from equation 3.4 is given explicitly by ?C?H = 1 and for h ??H by ?Ch = summationdisplay ???? e? ? ?D??A??h. Proof. As W?H = 1 also ?C?H = 1. Assume h ??H. Then u01(h???K) = summationdisplay i a?ih?epsilon1i ???K = summationdisplay i ?lscripti,h??H ?epsilon1i ???K + summationdisplay i ?a? i h?epsilon1i ???K. Because u?(?H ??K) = ?H ??P we obtain (with Lemma 4.1) for the first part limn??u?0n???u?02( summationdisplay i ?lscripti,h??H ?epsilon1i ???K) = summationdisplay i ?lscripti,h??H ?epsilon1i ???P = ?H ? ?D?h???P similarequal ?D?h???P ? ?P. Using the product formula from Theorem 1.1 and iterating the argument above we get ?C(h) = Wh = ??w??1A (h) = ??( ?D?h???P) + ?? limn??u?0n???u?02 summationdisplay i ?a? i h?epsilon1i ???K = e0 ? ?D?h + ?? limn??u?0n???u?03 summationdisplay j,i parenleftbig?lscript j, ?a? i h??H+ ?a? j ?a? i h parenrightbig?epsilon1 i ?epsilon1j ???K = e0 ? ?D?h+ dsummationdisplay i=1 ei ? ?D? ?a?i h + ?? limn??u?0n???u?03 summationdisplay j,i ?a? j ?a? i h?epsilon1i ?epsilon1j ?? ?K 14 Santanu Dey and Rolf Gohm = ... = summationdisplay |?|