A-Isometries and Hilbert-A-Modules Over Product Domains

For a compact set K⊂Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \subset {\mathbb {C}}^n$$\end{document}, let A⊂C(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \subset C(K)$$\end{document} be a function algebra containing the polynomials C[z1,⋯,zn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}[z_1,\cdots ,z_n ]$$\end{document}. Assuming that a certain regularity condition holds for A, we prove a commutant-lifting theorem for A-isometries that contains the known results for isometric subnormal tuples in its different variants as special cases, e.g., Mlak (Studia Math. 43(3): 219–233, 1972) and Athavale (J. Oper. Theory 23(2): 339–350, 1990; Rocky Mt. J. Math. 48(1): 2018; Complex Anal. Oper. Theory 2(3): 417–428, 2008; New York J. Math. 25: 934–948, 2019). In the context of Hilbert-A-modules, our result implies the existence of an extension map for hypo-Shilov-modules (i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(i=1,2)$$\end{document}. By standard arguments, we obtain an identification where is the minimal C(∂A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\partial _A)$$\end{document}-extension of (i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(i=1,2)$$\end{document}, provided that is projective and is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra A=A(D)=C(D¯)∩O(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A=A(D)=C({\overline{D}})\cap {\mathcal {O}}(D)$$\end{document} over a product domain D=D1×⋯×Dk⊂Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D = D_1 \times \cdots \times D_k \subset {\mathbb {C}}^n$$\end{document} where each factor Di\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_i$$\end{document} is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some Cdi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^{d_i}$$\end{document} (1≤i≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i \le k$$\end{document}). This extends known results from the ball and polydisc-case, Guo (Studia Math. 135(1): 1–12, 1999) and Chen and Guo (J. Oper. Theory 43: 69–81, 2000).


A-Isometries
Let H be a separable complex Hilbert space. A spherical isometry is a commuting tuple T = (T 1 , · · · , T n ) ∈ B(H) n of bounded linear operators on H that satisfies the condition T * 1 T 1 + · · · + T * n T n = 1 H . Athavale proved in [3] that spherical isometries are subnormal with normal spectrum σ n (T ) conained in ∂B n , a fact which provides them with a natural A(B n )-functional calculus, where A(B n ) = { f ∈ C(B n ) : f |B n is holomorphic} denotes the ball algebra. Replacing A(B n ) with a general function algebra A ⊂ C(K ) on a compact set K ⊂ C n and ∂B n with its Shilov-boundary ∂ A , one arrives at the notion of A-isometry as introduced by Eschmeier in [16] (see below for a precise definition). It is this general setting in which we will formulate most of our results. We use the same notations as in [14] and [17]. A reader familiar with one of these works may readily proceed with Sect. 2.

Subnormal tuples
A tuple T ∈ B(H) n is called subnormal, if there is a commuting tuple of normal operators U ∈ B( H) n on a Hilbert space H ⊃ H such that H is invariant for the components of U and T = U |H. We suppose in the following that U is minimal in the sense that the only reducing subspace for U containing H is H. The normal spectrum of T is defined as σ n (T ) = σ (U ), where σ (U ) denotes the Taylor spectrum of U . This is independent of the special choice of U . By a result of Putinar, the spectral inclusion σ n (T ) ⊂ σ (T ) holds. Spectral theory for normal tuples asserts the existence of a so-called scalar spectral measure, i. e., a finite positive Borel measure μ on σ (U ) with the property that there exists an isomorphism of von Neumann algebras U : L ∞ (μ) → W * (U ) ⊂ B( H) mapping z i → U i (i = 1, · · · , n), called the L ∞ -functional calculus of U . (Note that such a measure μ is unique up to mutual absolute continuity.) One then defines the restriction algebra recall that the L ∞ -calculus U satisfies a spectral mapping theorem of the form σ ( U ( f 1 ), · · · , U ( f k )) = F(σ (U )) for every k-tuple F = ( f 1 , · · · , f k ) ∈ C(K ) k with k ∈ N (see, e. g., [29], Section 0.3 and the references therein). A well-known uniqueness result for Taylor's functional calculus (cp. Theorem 5.2.4 in [18]) then guarantees that U ( f ) = f (U ) for every f ∈ O(W ) if W ⊂ C n is any open set containing σ (U ). If we even assume that W ⊃ σ (T ), we can apply Lemma 2.5.8 in [18] to the inclusion map i : H → H as intertwiner of T and U to deduce that f (U )|H = f (T ) for f ∈ O(W ). Hence the asserted uniqueness (1.1) follows.

A-subnormal tuples, A-isometries
Fix a compact set K ⊂ C n and let C(K ) denote the algebra of all continuous complexvalued functions equipped with the supremum norm. For a closed subalgebra A ⊂ C(K ), we write ∂ A for the Shilov boundary of A, i.e. the smallest closed subset of K such that f ∞,K = f ∞,∂ A . Following Eschmeier, [16], we define: If, in addition, σ n (T ) ⊂ ∂ A , then T is said to be an A-isometry. An A-isometry consisting of normal operators is called A-unitary.
Natural choices for A to consider are algebras of disc-algebra type, that is, where D ⊂ C n is a suitably chosen bounded open set. As pointed out above, setting A = A(B n ) yields exactly the class of spherical isometries, whereas A(D n )-isometries (D ⊂ C stands for the open unit disc) are precisely commuting tuples whose components are isometric operators on H.
Let now T ∈ B(H) n be an A-subnormal tuple for A ⊂ C(K ) as in the definition, and fix a scalar spectral measure μ of its minimal normal extension. Via trivial extension, we may regard μ as an element of M + (K ), the set of all finite regular Borel measures on K . Since R T is w * -closed and assumed to contain A, the restriction of γ T to the algebra is well defined. By the properties of γ T stated above, it therefore induces an isometric and w * -continuous isomorphism onto its image T a (T ) = γ T (H ∞ A (μ)) (the set of all so-called analytic T -Toeplitz operators). To keep the notation simple, we denote the induced map again by γ T , and call it the canonical H ∞ -functional calculus of T :

A-unitary dilations
Let again T ∈ B(H) n be A-subnormal with minimal normal extension U ∈ B( H) n . As a * -homomorphism, the map U |C(K ) : C(K ) → B( H) is completely contractive, and hence so is γ T |A : A → B(H), if A is regarded as an operator subspace of C(K ). Since both embeddings A ⊂ C(K ) and A ⊂ C(∂ A ) induce the same operator space structure on A (see [27,Theorem 3.9]), a theorem of Arveson (see, e.g., [27,Corollary 7.7]) asserts that there exists an A-unitary tuple V ∈ B(K) n on a Hilbert space K ⊃ H such that We call such a tuple an A-unitary dilation of T . When restricted to the smallest closed subspace that contains H and reduces V , it is said to be a minimal A-unitary dilation.

Regularity
If T ∈ B(H) n is an A-subnormal tuple, then one can use its canonical functional calculus γ T to carry over approximation properties of the underlying function algebra A to the algebra T a (T ) of analytic Toeplitz operators. As it turns out, of particular interest are approximation properties of A that are related to the so-called abstract inner function problem. In its classical form, the inner function problem asks if there are non-constant bounded analytic functions θ : B n → C on the unit ball B n such that their radial limitθ : ∂B n → C satisfies |θ| = 1 σ -a.e. (so-called inner functions).
Here, σ denotes the normalized surface measure on ∂B n . In his celebrated work [1], Aleksandrov succeded to solve the inner function problem for what he called "regular triples" (A, K , μ) (cf. Corollary 29 therein and Proposition 2.6 below for a suitable formulation in our context). The triple (A(B n ), B n , σ ) models the classical context. The precise definition of regularity reads as follows: Definition 2.1 Let K ⊂ C n compact and A ⊂ C(K ) a closed subalgebra.
(a) For μ ∈ M + (K ), the triple (A, K , μ) is called regular (in the sense of Aleksandrov Note that the condition on the support of μ in part (b) of the preceding definition results from the fact that the stated inclusion necessarily holds for regular triples. A concrete source of regular triples is the following embedding criterion due to Aleksandrov: We first recall some basic facts about bounded symmetric domains: By definition, a bounded domain D ⊂ C d is symmetric if, for each z ∈ D, there exists a biholomorphic map s z : D → D possessing z as isolated fixed point such that s z • s z = id D . We assume that D is circled at the origin, i. e., 0 ∈ D and z D ⊂ D for z ∈ C with |z| = 1. Every set of this type is convex [24,Corollary 4.6] and the Shilov boundary of A(D) consists precisely of those points in D with maximal Euclidean distance from the origin in C d [24,Theorem 6.5]. Since every finite product of bounded symmetric and circled domains is again of this type, we only have to consider one such factor in Theorem 2.3. To keep the notations in the proof simple, we restrict ourselves to the case where D 1 and D 2 are strictly pseudoconvex sets and D 3 is the circled and bounded symmetric factor. This is no restriction, since the result is well-known for k = 1 (see the references given above), whereas the cases k = 2 and k > 3 can be handled by straight-forward modifications of our construction described below. In the following, we write S D = ∂ A(D) .
To start with, let us fix real numbers r 1 , r 2 , r 3 > 0 such that We first calculate the Shilov boundary of D. From the existence of peaking functions for every boundary point of a strictly pseudoconvex domain (see Range [28, Corollary VI.1.14]), it follows that S D i = ∂ D i for the strictly pseudoconvex factors i = 1, 2.
Repeating the argument twice, we obtain w 2 ∈ ∂ D 2 and in the latter product set. To finish the proof, it suffices to observe that h(z 1 , z 2 , is a peaking functions for w i (i = 1, 2, 3). For the strictly pseudoconvex factors, such functions exist by the reference given above. For D 3 , we may take the restriction of a peaking function for w 3 The key ingredient in the proof of Theorem 2.3, is an embedding theorem of Løw [25,Theorem 3]. Reduced to what we need, this theorem says the following: Given a strictly pseudoconvex domain D ⊂ C n with C 2 -boundary and a strictly positive continuous function φ : ∂ D → (0, ∞), there exists a mapping g ∈ A(D) l for some natural number l such that |g(z)| = φ(z) for every z ∈ ∂ D.
Proof of Theorem 2. 3 Since, by assumption, are strictly positive if | · | denotes the Euclidean norm on C d i . By the cited theorem of Løw, we find Then we set r = r 2 1 + r 2 2 + r 2 3 and define Together with Lemma 2.4 and the fact that S D 3 ⊂ r 3 · ∂B d 3 , this yields the inclusion F(S D ) ⊂ ∂B m . The desired regularity now follows as an application of Aleksandrov's embedding criterion stated as Theorem 2.2 above.

Remark 2.5
Let F ∈ A(D) m , and let | · | denote the Euclidean norm and ·, · the Euclidean scalar-product on C m . By considering scalar-valued functions of the form F(·), F(w) with suitably chosen w ∈ D, one can show that |F| takes its maximum value on the Shilov boundary S D , and that F must be constant if |F| takes its maximum at some point inside D. In view if this, we can -for further reference -state some additional properties of the map F constructed in the proof of Theorem 2.3, namely F(D) ⊂ B m and F(D) ⊂ B m . Moreover, we remark that the map F : D → F(D) is a homeomorphism, as it is a continuous bijection from a compact space onto a Hausdorff space.
As for the scope of Theorem 2.3, we should remark that every bounded open set D ⊂ C with C 2 boundary is strictly pseudoconvex, cp. the remark at the end of Section 1.5 in [22]. So, in particular A(D) ⊂ C(D), is regular for polydomains Now, let (A, K , μ) be a regular triple. In analogy with the classical case we define a μ-inner function to be an element of the set Our study of A-isometries in the following sections relies on a density result for such abstract inner functions taken from [14]. It is a slight variation of Aleksandrov's corresponding result [1,Corollary 29]. In its original form, it says that, given the regularity of (A, K , μ), the w * -closure of the set I μ in L ∞ (μ) contains the set Continuity means in this context that every one-point set has μ-measure zero. The key point here is that the following result holds without any continuity assumption on μ.
Note that, in the context of the preceding proposition, the first of these density relations implies the second one: The w * -closed linear hull on the right is easily seen to be a w * -closed subalgebra of L ∞ (μ) that is closed under complex conjugation. Since it contains the restrictions of all polynomials (by the first density relation), it actually must be all of L ∞ (μ). It has turned out (cp. [14,17,19]) that this kind of density is exactly what is needed for operator-theoretic applications. So we define:

Commutant Lifting for A-Isometries
The aim of this section is to prove the following commutant-lifting theorem that applies by the preceding section in particular to all algebras of the form A(D 1 × · · · × D n ) where each of the factors D i is either a bounded strictly pseudoconvex domain with C 2 -boundary, or a bounded sysmmetric and circled domain.
possesses a unique extension to an operator X ∈ B( H 1 , H 2 ) with the property that This extension satisfies X = X . If X has dense range, then so has X. If U 1 is even an A-unitary extension of T 1 and, in addition, X is isometric, then so is X.
The above theorem contains several known intertwining results for subnormal isometric tuples as special cases (note that all but the last one require T 1 to be an A-isometry): • A result of Guo (see [21,Lemma 3.6]) from the context of normal Hilbert modules.
Actually our proof is inspired by Guo's idea. But his hypotheses are rather special: Roughly speaking, both tuples are assumed to be spherical isometries, and the spectral measures of their minimal normal extensions are assumed to be absolutely continuous with respect to the surface measure of the sphere. • An intertwining result for spherical isometries by Athavale (Proposition 8 in [3]).
More generally, the analogous result for so called ∂ D-isometries introduced by Athavale in [4], where D ⊂ C n is a relatively compact strictly pseudoconvex domain: Our Theorem 3.1 implies Athavale's Theorem 3.2.
We carry out the details for the convenience of the reader: First of all, note that In Athavales result, two ∂ D-isometries T k = (T k,1 , · · · , T k,n ) ∈ B(H k ) n (k = 1, 2) are assumed to satisfy the inclusion σ (T 1 ) ∪ σ (T 2 ) ⊂ and an intertwining relation of the form To apply our theorem, it remains to check that hold in this case. But the assumed intertwining relation for the components immediately implies that X f ( is built using the holomorphic functional calculus. Again, the uniqueness of the canonical H ∞ -calculi of T 1 and T 2 then implies that For the proof of Theorem 3.1, we need an operator-theoretic version of the measuretheoretic density assumption from Definition 2.7. To formulate it appropriately, let us fix the following notation: Given two measures μ, Clearly, every r ν μ is a w * -continuous and contractive *-homomorphism. Given an A-unitary tuple U ∈ B( H) n with scalar-valued spectral measure μ ν, we denote by ν U the composition As a consequence of Proposition 2.6, we have: into itself. By linearity and continuity, we may pass to L H, so Since invariant subspaces are preserved under w * -limits, the w * -regularity assumption (cf. Definition 2.7) allows us to conclude that ν . By taking f to be z i and z i (i = 1, · · · , n), we see that H 0 is actually reducing for U . Standard arguments based on the theorems of Stone-Weierstraß and Lusin yield the stated minimality.
Besides the above density result, an essential ingredient of the proof is the simple observation that, for every θ ∈ I ν , the operator ν U (θ ) is isometric, since it evidently satisfies ν U (θ ) * ν U (θ ) = ν U (θθ) = 1 H . (Actually, it is even unitary as θ is the inverse of θ in L ∞ (ν), but isometry will suffice for our purposes.) Proof of Theorem 3.1 For k = 1, 2, let μ k ∈ M + (K ) denote the trivial extensions of scalar spectral measures of U k to elements in M + (K ). The proof will actually work if we replace the hypothesis on A to be a w * -regular function algebra with the following weaker but more technical one: There exists a measure ν ∈ M + (∂ A ) such that (A, K , ν) is w * -regular and μ 1 , μ 2 ν. Let us fix such a measure ν for the rest of the proof. (Under the original hypothesis of the theorem, ν = μ 1 + μ 2 evidently has the desired properties.) Throughout the proof, we use the abbreviations We first define X on a dense subset of H 1 (cf. Lemma 3.2) in the obvious way, namely where F ⊂ N denotes a finite set and θ i , η i ∈ I ν , h i ∈ H are arbitrary elements for i ∈ F. To see that this is well defined, we estimate the norm of the right-hand side. If we define π = i∈F θ i , then π ∈ I ν , and so the operator ν U 2 (π ) is isometric. This yields From the very definition of π , the functions π θ i belong to I ν for every choice of i ∈ F. Since, in addition, X maps into H 2 , and U 2 is an A-unitary extension of T 2 , we may replace ν U 2 with γ ν T 2 . Then we use the intertwining relation A) from the hypothesis. We restate this as to see that it extends by w * -WOT-continuity to These considerations allow us to continue expression (3.2) as follows where we again used the isometry of ν U 1 (π ) in the last step. First of all, this shows that X is indeed well defined (and then, by construction, linear) on the space F of all finite linear combinations of vectors of the form ν In particular, X maps Cauchy-sequences in F ⊂ H 1 to Cauchy-sequences in H 2 and thus can be continuously extended to a linear map on the closure F, which coincides with H 1 by Lemma 3.2. This map has obviuously norm less than X , and a look at the assumed density (Definition 2.7 (b)) and the defining formula (3.1) for X shows that it actually has the desired intertwining property. In view of Lemma 3.2 and (3.1), X has dense range if so has X . Moreover, if U 1 is even an A-unitary extension of T 1 , then we may drop the operator P H in the considerations preceding estimate (3.3). Thus, if X is isometric, then equality holds in (3.3) and (3.4) with X = 1.
To finish the proof, note that an operator X satisfying the intertwining relation from the statement of the theorem necessarily fulfills X ν Consequently, it satisfies (3.1), and therefore coincides with the extension constructed above.
When specialized to one A-isometry, Theorem 3.1 reads as follows:

Lifting of Module Homomorphisms
Let A ⊂ C(K ) be function algebra, i. e., a closed subalgebra of C(K ) that separates the points of K . Recall that a Hilbert-A-module is a Hilbert space H together with a continuous bilinear map  The key observation for applying Theorem 3.1 in the context of Hilbert modules is the following simple lemma which says that A-isometries and Shilov-A-modules are essentially the same.  By an A-module homomorphism X : H → K between two Hilbert-A-modules we mean a bounded linear map that respects the module multiplication in the sense that X ( f · h) = f · Xh whenever f ∈ A and h ∈ H. We write Hom A (H, K) to denote the set of all A-module homomorphisms from H to K.

(a) If S ⊂ H is a Shilov-A-module with minimal contractive C(∂ A )-extension H, then M z ∈ B(S) n is an A-isometry with minimal normal extension M z ∈ B(H) n . (b) Conversely, if T ∈ B(S) n is an A-isometry with minimal normal extension U ∈ B(H) n , then, introducing the multiplications A
Let now two Shilov modules S 1 , S 2 with minimal C(∂ A )-extensions H 1 , H 2 be given. Using the identifications from the preceding proof, namely C(∂ A )), the following identities are evident: As a last ingredient for the extension theorem we aim for (Theorem 4.4), we need the following well-known fact: Now we are ready to prove the main result of this section. In the case A = A(B n ), this appears as Lemma 3.6 in [21] (where it is stated for Shilov-modules only, and under an additional continuity assumption). For a unit modulus algebra A, the corresponding result appears as Proposition 2.5 in [9]. Our theorem contains both as special cases.

Theorem 4.4 Let S k be hypo-Shilov modules over A with C(∂ A )-extensions H
If H 1 is minimal, then ε is unique.
Proof The proof is divided into three steps.
Step I: Reduction to the minimal case. To prove the existence of ε, we may assume that H k are minimal C(∂ A )-extensions of S k (k = 1, 2). To see this, let K k are arbitrary C(∂ A )-extensions of S k , and set H k = C(∂ A ) · K k S k , which is minimal. Then H 1 is a reducing submodule of K 1 , whence the projection P H 1 ∈ B(K 1 ) is a C(∂ A )module homomorphism, as is the inclusion map i K 2 : H 2 → K 2 (for trivial reasons). Assuming the existence-assertion of the theorem to hold for the minimal extensions H k , we may set to obtain an extension mapping ε : in the general case. To finish the first step, we have to justify that replacing K 2 with H 2 does not impose any restriction on the uniqueness assertion either: But if H 1 is assumed to be minimal (as it is in the uniqueness part), then every X ∈ Hom C(∂ A ) (H 1 , K 2 ) satisfies X (C(∂ A ) · S 1 ) ⊂ C(∂ A ) · S 2 and thus has range in H 2 = C(∂ A ) · S 2 , and may actually considered an element of Hom C(∂ A ) (H 1 , H 2 ). So in what follows, we may and shall assume H k to be a minimal extension of S k (k = 1, 2).
Step II: Reduction to the Shilov-case. According to Theorem 4.3, we find invertible Hilbert-C(∂ A )-module maps onto contractive Hilbert-C(∂ A )-modules. Since restrictions S k |S k : S k → S k S k are also invertible A-module maps, the vertical maps in the following diagram which are defined as "conjugations" are bijections. Assuming the theorem to hold for Shilov-modules and their minimal extensions, there is an extension map ε S from Hom A (S 1 S 1 , S 2 S 2 ) to Hom C(∂ A ) (K 1 , K 2 ) as stated in the theorem, which we take as the lower horizontal map of the diagram. We define the upper horizontal map so to make the diagram commutative. To state it explicitly, The stated extension property ε(X )|S 1 = X for X ∈ Hom A (S 1 , S 2 ) is obvious. Moreover, using the diagram, it is easy to see that ε is unique with the stated extension property if and only if so is ε S with the corresponding property.
Step III: The Shilov-case with minimal extensions. So we may finally assume that S k are Shilov-modules with minimal C(∂ A )-extensions H k for k = 1, 2. By Lemma 4.2 (a) and the subsequent remarks on A-module homomorphisms, the statement of the theorem is nothing but a reformulation of Theorem 3.1. Thus, the proof is complete.
The existence of a lifting map of the above type implies by standard arguments the existence of an "Hom-isomorphism theorem" for hypo-Shilov-modules and their complements (cp. [21] and [9]). To give an appropriate formulation (see Theorem 6.1 below), we need the concept of projectivity, which is introduced and studied for A(D)modules over product domains in the next section.
1 . We refer to [7] for the homological background. Proposition 2.1.5 therein relates the characterization of injective and projective objects to the vanishing of cohomology groups Ext 1 H A (K, H) defined as follows: Two short exact sequences commutative. The first cohomology group is then defined by The zero element of Ext 1 where i H and P K denote the canonical inclusion and projection, respectively. In terms of Ext-groups, an element K ∈ H A is projective, if and only if Ext 1 H A (K, H) = 0 for every H ∈ H A , and injective, if and only if Ext 1 H A (H, K) = 0 for every H ∈ H A . It was shown by Carlson and Clark [8] for D = D n , by Guo [21] for D = B n (under an additional continuity assumption on the underlying modules related to the surface measure on ∂B n ) and by Eschmeier and the author [13]  To prove this, we basically use an emedding argument similar to the one used to deduce the bounded symmetric case from the strictly pseudoconvex one in [13] (see the remarks preceding Corollary 3.2 therein). While the embedding used in the latter case was the trivial one (namely, the inclusion of a bounded symmetric and circled domain into its envelopping ball), the construction of a suitable embedding is the non-trivial part in the product setting under consideration.
In preparation of the main construction we show the following auxiliary lemma: Proof It is elementary to check that S, viewed as a map A( is a well-defined isometry. To see that it actually maps into A(D 2 , A(D 1 )), we have to make sure that S f |D 2 is analytic as a function with values in A(D 1 ). Since A(D 1 ) isometrically embeds as a subspace into H ∞ (D 1 ) (by restriction), it suffices to check the holomorphy of S f |D 2 , viewed as a map D 2 → H ∞ (D 1 ). But this follows from [12,Lemma 5.4], which settles the corresponding identification ). So the map S from the statement of the theorem is actually a well-defined isometry. To conclude the proof, it suffices to check that S is surjective.
Towards this, let f ∈ A(D 2 , A(D 1 )) be given. Applying a suitable translation, we may assume that D 2 contains the origin. Using the uniform continuity of f , one shows that, for r ↑ 1 (0 < r < 1), the functions f r defined by f r (z) = f (r z) which are holomorphic on the open set D r = (1/r ) · D 2 ⊃ D 2 converge to f uniformly on D 2 . Thus we can choose a sequence (r n ) in such a way that the functions In view of the well known identification O(D r n ) ⊗A(D 1 ) ∼ = O(D r n , A(D 1 )) (see, e. g., [30]), where the latter space carries the topology of uniform convergence on compact subsets, we can choose, for each n ≥ 1, a natural number k n and functions In a final approximation step, we make use of the (polynomial) convexity of D 2 and use the Oka-Weil theorem to find polynomials ( p n i ) 1≤i≤k n in C[z] such that for each fixed n ≥ 1. By construction, we have To finish the proof, it suffices to observe that the functions g n i defined by g n i (z 1 , and are mapped to p n i ⊗ h n i via S. This shows that S has dense range and thus is surjective, as desired.
The previous lemma helps us to handle the symmetric factor in the following embedding theorem. The strictly pseudoconvex parts will be handled by combining the embedding theorems of Fornaess [20,Theorem 9] and Løw [25] with a result of Jacóbczak [23, Theorem 1] on the existence of extension operators.

Proposition 5.3
Let D = D 1 ×· · ·× D k ⊂ C n be a product domain as in the statement of Theorem 5.1. Then there exist constants N ∈ N, r > 0 and a mapping F ∈ A(D) N such that, with B = r · B N , the following assertions hold: Proof As in the proof of Theorem 2.3, we may restrict ourselves to the case where D 1 and D 2 are strictly pseudoconvex factors and D 3 is a bounded symmetric and circled open set. The proof is divided into three steps: Step I: Reduction to the case of convex domains. All of the following assertions hold for i = 1, 2: Since D i is a Stein compactum and hence holomorphically convex, we may apply Fornaess' embedding theorem [20,Theorem 9] (in the special case X = C d i ), to obtain holomorphic maps ψ i : C d i → C m i and strictly convex bounded domains C i ⊂ C m i with C 5 -boundary such that If we set M i = ψ i (D i ) and note that M i = M i ∩ C i (with transversal intersection at ∂C i ), then these sets fulfill exactly the hypotheses of [23, Theorem 1] (cp. also the Remark preceding Section 4 therein). We write ψ = (ψ 1 , ψ 2 ) : D 1 × D 2 → M 1 × M 2 and consider the following commuting diagram • the vertical maps are given by the identification established in the previous lemma, • the mapping 1 ⊗ C ψ acts by pointwise application of the composition operator • the upper horizontal map R (which we think of as a restriction map) is by definition the composition operator with the map (ψ, id) : It is elementary to check that the diagram commutes. We claim that R is surjective, which lifts our problem to a product of convex sets (in the upper left corner). Since 1 ⊗ C ψ is obviously a topological isomorphism (with inverse 1 ⊗ C ψ −1 ), the difficulty lies in checking that 1 ⊗ R M is surjective. By a result of Jacóbczak [23,Theorem 1] (and the remark at the end of Section 3 therein), there is a continuous linear extension operator )-functions, we obtain a bounded linear right inverse for 1 ⊗ R M , and the claimed surjectivity follows.
Step II: Embedding a product of convex domains into a ball. Towards an application of Løw's embedding theorem to the sets C i as in the proof of Theorem 2.3 on p. 4, Let us choose r i > 0 such that C i ⊂ r i · B m i (i = 1, 2) and D 3 ⊂ r 3 · B d 3 with Then, exactly as in the cited proof, the application of Løw's theorem yields a map with g i ∈ A(C i ) l i for suitable l i ∈ N, such that with r = r 2 1 + r 2 2 + r 2 3 and B = r · B N with N = m 1 + m 2 + d 3 + l 1 + l 2 the following assertions hold (consider Remark 2.5): To argue as in the referenced proof, note that the sets C i are strictly pseudoconvex (as they are strictly convex, cp. p. 529 of [20] To finally obtain a map F ∈ A(D) N as in the statement of the theorem, we set Properties (a) and (b) from the assertion of the theorem are valid by construction, so it remains to check that (c) holds.
Step III: The range of C F : where C G ( f ) = f • G and R is the upper horizontal map from (5.1). To complete the proof, it suffices to check that C G : A(B) → A(C 1 ×C 2 ×D 3 ) has dense range. Assuming that, for a moment, it follows immediately that C F (A(B)) ⊃ R(C G (A(B))) = R(A(C 1 × C 2 × D 3 )) = A(D), since R is surjective as we have seen in Step I. So let us finally aim for the density of C G . After a suitable translation, we may of course assume that the set C 1 × C 2 × D 3 ⊂ C m contains the origin. Using radial approximation and the fact that the set C 1 × C 2 × D 3 is polynomially convex, it suffices to check that the range of C G contains the polynomials in z = (z 1 , · · · , z m ). Since C G is a unital algebra homomorphism, it suffices to check that the range of C G contains the (restrictions) of the coordinate projections π 1 , · · · , π m : C m → C, i. e., π i (z) = z i . But, by construction (see (5.2) above), G acts as the identity on the first m components, so if we denote the coordinate functions on C N with π N j : C N → C, then we clearly have π N j |B ∈ A(B) and (π N j |B) • G(z) = z j = π j (z) for j = 1, · · · , m and z = (z 1 , · · · , z m ) This observation finishes the proof.  ) . Towards this, let F : D → B be the embedding constructed in Proposition 5.3. Note that H, K and J can be viewed as A(B)-modules when we introduce the module multipliation analogously for K, J. Since F(S D ) ⊂ ∂ B, this module structure extends to a C(S D )module structure for H. As the Ext 1 -group is known to vanish in the ball case by

A Hom-Isomorphism Theorem
Let H be a Hilbert-A-module and S ⊂ H a closed A-submodule, that is, a closed subspace satisfying M H f S ⊂ S for f ∈ A. If we write P S , P S ⊥ ∈ B(H) for the orthogonal projections with ranges S and S ⊥ , respectively, then the identity valid for arbitrary f , g ∈ A shows that S ⊥ carries a natural A-module structure, where the module multiplication is given by Furthermore, the space Hom A (H 1 , H 2 ) can be equipped with an A-module structure by setting f · X = X • M H 1 f ( f ∈ A, X ∈ Hom A (H 1 , H 2 )). Proof For the convenience of the reader, we repeat the known argument from the proof of [21,Theorem 3.7]. The map β from the statement of the theorem is defined in such a way that the following diagram commutes: Only the bijectivity of β requires an argument. To see that β is injective, suppose that β(X ) = 0 for some X ∈ Hom A (S 1 , S 2 ). It follows immediately that ε(X )H 1 ⊂ S 2 is a reducing submodule. Since S 2 is assumed to be pure, we conclude that ε(X ) = 0 and thus X = 0, as desired. For the surjectivity, let an arbitrary Y ∈ Hom A (S ⊥ 1 , S ⊥ 2 ) be given. By the assumed projectvity of H 1 , we may choose an A-module homomorphism X : H 1 → H 2 such that P S ⊥ 2 X = Y P S ⊥ 1 . Setting X = X |S 1 , it follows that ε(X ) = X from uniqueness, and so β(X ) = P S ⊥ 2 X |S ⊥ 1 = Y .
The limiting factor for an application of the preceding theorem is the projectivity of H 1 . In view of the fact that every C(S D )-extension of a Hilbert-A(D)-module is projective for D ⊂ C n if D is either (i) a bounded strictly pseudoconvex set or (ii) a circled bounded symmetric domain (see [13,Theorem 3.1]) or (iii) a product of finitely many such domains (Theorem 5.1), the above result allows us to extend Guo [21,Theorem 3.7] and Chen and Guo [9, Theorem 3.5] to these kinds of sets. In particular, the restriction to Shilov-modules and the additional continuity assumption made in [21] to handle the ball case are not necessary. Assume that S i are hypo-Shilov modules with minimal C(S D )-extensions H i (i = 1, 2) and that H 2 is pure. Then we have an isomorphism β : Hom A(D) (S 1 , S 2 ) → Hom A(D) (H 1 , H 2 ) as stated in the above theorem.
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