Let \(\Delta \) be a closed, cocompact subgroup of \(G \times \widehat{G}\), where G is a second countable, locally compact abelian group. Using localization of Hilbert \(C^*\)-modules, we show that the Heisenberg module \(\mathcal {E}_{\Delta }(G)\) over the twisted group \(C^*\)-algebra \(C^*(\Delta ,c)\) due to Rieffel can be continuously and densely embedded into the Hilbert space \(L^2(G)\). This allows us to characterize a finite set of generators for \(\mathcal {E}_{\Delta }(G)\) as exactly the generators of multi-window (continuous) Gabor frames over \(\Delta \), a result which was previously known only for a dense subspace of \(\mathcal {E}_{\Delta }(G)\). We show that \(\mathcal {E}_{\Delta }(G)\) as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if \(\Delta \) is a lattice, and their associated frame operators corresponding to \(\Delta \) are bounded.

中文翻译：

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海森堡模块作为函数空间

假设\（\ Delta \）是\（G \ times \ widehat {G} \）的一个封闭的，紧紧的子群，其中G是第二个可数的局部紧的阿贝尔群。使用希尔伯特\（C ^ * \）-模块的本地化，我们证明了在扭曲组\（C ^ * \）-代数上的Heisenberg模块\（\数学{E} _ {\ Delta}（G）\） Rieffel产生的\（C ^ *（\ Delta，c）\）可以连续且密集地嵌入希尔伯特空间\（L ^ 2（G）\）中。这使我们能够表征为一组有限的发电机\（\ mathcal {E} _ {\德尔塔}（G）\）的多窗口的完全发电机（连续）的Gabor帧超过\（\德尔塔\），以前只对\（\ mathcal {E} _ {\ Delta}（G）\）的密集子空间知道的结果。我们证明\（\ mathcal {E} _ {\ Delta}（G）\）作为函数空间满足两个属性，使其有资格进行时频分析：如果\（\ Delta \）是一个格，并且它们与\（\ Delta \）对应的关联帧运算符是有界的。