Lie Semigroup Operator Algebras.
Lie Semigroup Operator Algebras.
复制标题
李半群算子代数。
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
R. Levene
中科院分区：
文献类型：
--
作者：
R. Levene
抛物线代数 Ap 和双曲代数 Ah 是非自共轭 w* 闭算子代数，首先由 A. Katavolos 和 S. C. Power 考虑。在[KP97]和[KP02]中，他们证明了它们的不变子空间格同胚于紧连通欧几里得流形，并且抛物线代数在哈尔莫斯意义上是自反的。通过对希尔伯特-施密特算子的分析，我们给出了抛物代数自反性的新证明。我们还证明了 Ap 中存在具有非平凡核的算子。然后，我们考虑抛物线代数的一些自然“伴生代数”，这导致了称为傅立叶-普朗舍雷球的紧凑子空间晶格。我们证明该晶格的酉自同构群同构于 R2 和 SL2(R) 的半直积。双曲代数具有自反性的证明是通过对希尔伯特-施密特算子进行与用于建立 Ap 的自反性的分析基本相同的分析得出的。我们还提供了有关投影的强算子拓扑限制的已知结果的透明证明。两个 Katavolos-Power 代数都是由酉值表示下李群的半群的图像生成为 w* 闭算子代数。按照[KP02]，我们将此类算子代数称为李半群算子代数。我们通过考虑 SL2(R) 酉值表示下李群 SL2(R) 的半群 SL2(R+) 的图像来寻找此类代数的新例子。我们证明以这种方式产生的特定李半群算子代数 A+ 是自反的，并且它是留下双三角形子空间晶格不变量的算子代数。令人惊讶的是，A+ 是由 SL2(R+) 的真子半群的图像生成的 w* 闭代数。
The parabolic algebra Ap and the hyperbolic algebra Ah are nonselfadjoint w*-closed operator algebras which were first considered by A. Katavolos and S. C. Power. In [KP97] and [KP02] they showed that their invariant subspace lattices are homeomorphic to compact connected Euclidean manifolds, and that the parabolic algebra is reflexive in the sense of Halmos. We give a new proof of the reflexivity of the parabolic algebra through analysis of Hilbert-Schmidt operators. We also show that there are operators in Ap with nontrivial kernel. We then consider some natural "companion algebras" of the parabolic algebra which leads to a compact subspace lattice known as the Fourier-Plancherel sphere. We show that the unitary automorphism group of this lattice is isomorphic to a semidirect product of R2 and SL2(R). A proof that the hyperbolic algebra is reflexive follows by an essentially identical analysis of Hilbert-Schmidt operators to that which was used to establish the reflexivity of Ap. We also present a transparent proof of a known result concerning a strong operator topology limit of projections. Both of the Katavolos-Power algebras are generated as w*-closed operator algebras by the image of a semigroup of a Lie group under a unitary-valued representation. Following [KP02], we call such operator algebras Lie semigroup operator algebras. We seek new examples of such algebras by considering the images of the semigroup SL2(R+) of the Lie group SL2(R) under unitary-valued representations of SL2 (R). We show that a particular Lie semigroup operator algebra A+ arising in this way is reflexive and that it is the operator algebra leaving a double triangle subspace lattice invariant. Surprisingly, A+ is generated as a w*-closed algebra by the image of a proper subsemigroup of SL2(R+).