Let $F$ be a right Hilbert $C$*-module over a $C$*-algebra $B$, and suppose that $F$ is equipped with a left action, by compact operators, of a second $C$*-algebra $A$. Tensor product with $F$ gives a functor from Hilbert $C$*-modules over $A$ to Hilbert $C$*-modules over $B$. We prove that under certain conditions (which are always satisfied if, for instance, $A$ is nuclear), the image of this functor can be described in terms of coactions of a certain coalgebra canonically associated to $F$. We then discuss several examples that fit into this framework: parabolic induction of tempered group representations; Hermitian connections on Hilbert $C$*-modules; Fourier (co)algebras of compact groups; and the maximal $C$*-dilation of operator modules over non-self-adjoint operator algebras.

中文翻译：

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希尔伯特$ C $ *-modules的下降

假设$ F $是$ C $ *-代数$ B $的右希尔伯特$ C $ *-模块，并假定$ F $配备了紧致运算符向左移动的第二个C $ * -代数$ A $。具有$ F $的Tensor产品提供了从$ A $的Hilbert $ C $ *模块到$ B $的Hilbert $ C $ *模块的函子。我们证明，在某些条件下（例如，如果$ A $是核能，则总是可以满足），该函子的图像可以用与$ F $正则相关的某些定居代数的互作用来描述。然后，我们讨论适合该框架的几个示例：抛物线归纳的钢化群体表示；希尔伯特$ C $ *-模块上的Hermitian连接；紧群的傅立叶（共）代数；和非自伴算子代数上算子模块的最大$ C $ *扩张。