engleski

Topologically finitely generated Hilbert C(X)-modules

For a Hilbert $C(X)$-module $V$, where $X$ is a compact metrizable space, we show that the following conditions are equivalent: (i) $V$ is topologically finitely generated, (ii) there exists $K \in \N$ such that every algebraically finitely generated submodule of $V$ can be generated with $k \leq K$ generators, (iii) $V$ is canonically isomorphic to the Hilbert $C(X)$-module $\Gamma(\mathcal{; ; ; E}; ; ; )$ of all continuous sections of an (F) Hilbert bundle $\mathcal{; ; ; E}; ; ; =(p, E, X)$ over $X$, whose fibres $E_x$ have uniformly finite dimensions, and each restriction bundle of $\mathcal{; ; ; E}; ; ; $ over a set where $\dim E_x$ is constant is of finite type, (iv) there exists $N \in \N$ such that for every Banach $C(X)$-module $W$, each tensor in the $C(X)$-projective tensor product $V \po_{; ; ; C(X)}; ; ; W$ is of (finite) rank at most $N$.

Hilbert C(X)-module ; (F) Hilbert bundle ; Subhomogeneous ; Finite type property ; C(X)-projective tensor product

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