The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on \(L^2({{\mathbb {R}}}^\times , dt/ |t|)\). In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on \(L^p({{\mathbb {R}}}^\times , dt/ |t|)\) for \(p\in (1,2)\cup (2,\infty )\).

实线的仿射群的傅立叶代数具有自然的标识，即Banach空间，具有\（L ^ 2（{{\\ mathbb {R}}} ^ \ times，dt上的跟踪类算符空间/ | t |）\）。在本文中，我们研究了与傅立叶代数中的点乘积相对应的跟踪类算子的“双卷积积”。回答了Eymard和Terp工作中提出的问题，我们对该操作进行了内在描述，该操作不依赖于傅立叶代数的标识，并且对该仿射组的连接部分获得了相似的结果。在这两种情况下，我们都在相应的Banach代数上构造显式推导，直接验证推导恒等式而无需逆傅立叶变换。我们还针对\（p \ in ）在\（L ^ p（{{\ mathbb {R}}} ^ \ times，dt / | t |）\）上针对跟踪类算子的类似Banach代数结构的研究开始（1,2）\ cup（2，\ infty）\）。