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

For a compact set $$K \subset {\mathbb {C}}^n$$ K ⊂ C n , let $$A \subset C(K)$$ A ⊂ C ( K ) be a function algebra containing the polynomials $${\mathbb {C}}[z_1,\cdots ,z_n ]$$ C [ z 1 , ⋯ , z n ] . 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; NewYork J. Math. 25: 934–948, 2019). In the context of Hilbert-A-modules, our result implies the existence of an extension map ε : HomA(S1, S2) → HomC(∂A)(H1,H2) for hypo-Shilov-modules Si ⊂ Hi $$(i=1,2)$$ ( i = 1 , 2 ) . By standard arguments, we obtain an identification HomA(S1, S2) ∼= HomA(H1 _ S1,H2 _ S2) where Hi is the minimal $$C(\partial _A)$$ C ( ∂ A ) -extension of Si, $$(i=1,2)$$ ( i = 1 , 2 ) provided thatH1 is projective and S2 is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra $$A=A(D)=C({\overline{D}})\cap {\mathcal {O}}(D)$$ A = A ( D ) = C ( D ¯ ) ∩ O ( D ) over a product domain $$D = D_1 \times \cdots \times D_k \subset {\mathbb {C}}^n$$ D = D 1 × ⋯ × D k ⊂ C n where each factor $$D_i$$ D i is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some $${\mathbb {C}}^{d_i}$$ C d i ( $$1\le i \le k$$ 1 ≤ i ≤ k ). 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).