AN EXTENSION OF PLANCHEREL'S FORMULA TO SEPARABLE UNIMODULAR GROUPS
AN EXTENSION OF PLANCHEREL'S FORMULA TO SEPARABLE UNIMODULAR GROUPS
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DOI：
10.2307/1969470
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发表时间：
1950-09
影响因子：
4.9
通讯作者：
I. Segal
中科院分区：
文献类型：
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作者：
I. Segal
证明了对于可分么模局部紧群G(相对于Haar测度)上的任意函数f平方可积，f的绝对值的平方在G上的积分等于f的傅里叶变换的相对范数(在空间的每一点)的平方在某一度量空间上的积分。我们所说的“相对范数”指的是von Neumann[5]定义的因子相对范数，目前我们利用von Neumann约化理论[6]定义了傅立叶变换。本文得到的公式有Plancerel公式，它推广到可分局部紧交换群，Peter-Weyl定理，以及Gelfand和Neumark[2]最近针对Lorentz群得到的一个公式作为例子，但不是直接推论。所讨论的测度空间具有同构于L2(G)中的闭线性流形的布尔环B的测度环，该布尔环在左右平移下都是不变的，并且相应的测度是唯一确定的，即空间中每一点的相对范数的模归一化.这个测度环对群来说是一种测度论对偶。例如，对相对范数进行所谓的“标准”正规化，如果G是交换的，则B作为测度环在Haar测度下抽象地等同于G的特征标群的测度环；如果G是紧的，则B抽象地是离散点集的测度环，该集合与G的连续不可约表示的等价类集合一一对应，点的度量与相应表示的次数成比例。我们工作中的基础是某种可数可加的非负函数，我们称之为G的“对偶规”，它定义在L2(G)中左平移生成的代数的弱闭包V中的所有投影格上，或者定义在L2(G)中右平移不变的闭线性流形的格上。对偶规范以一种内在的方式定义，在V中的么正算子的变换下是不变的，实际上是冯·诺依曼[6]术语中的“权函数”。因此，他的约化理论将对偶规表示为上述由因素引起的成分(相对维度函数)在度量空间上的积分。这个表示，再加上L2(G)上由L2(G)的自伴元定义的卷积算子是超极大对称的(Ambrose[1])，以及L2(G)上每一个与所有右平移可交换的有界线性算子都在V中(西格尔[4])，是我们推导广义Plancerel公式所用的主要已知结果。
We show that for any function f square-integrable on a separable unimodular locally compact group G (relative to Haar measure), the integral over G of the square of the absolute value of f equals the integral over a certain measure space of the square of the relative norm (at each point of the space) of the Fourier transform of f. By "relative norm" we mean that defined by von Neumann [5] for factors, and for the present we define the Fourier transform thru the use of the von Neumann reduction theory [6]. The formula which is obtained here has as instances, though not as direct corollaries, the formula of Plancherel, its generalization to separable locally compact abelian groups, the Peter-Weyl theorem, and a formula recently obtained by Gelfand and Neumark [2] for the case of the Lorentz group. The measure space in question has its measure ring isomorphic to the Boolean ring B of closed linear manifolds in L2(G) invariant under both left and right translations, and the corresponding measure is uniquely determined, modulo normalization of the relative norm at each point of the space. This measure ring acts as a kind of measure-theoretic dual to the group. For example, making the so-called "standard" normalization of the relative norm, if G is abelian, then B as a measure ring is abstractly identical with the measure ring of the character group of G, under Haar measure; if G is compact, then B is abstractly the measure ring of a discrete set of points, the set being in one-to-one correspondence with the collection of equivalence classes of continuous irreducible representations of G, the measure of a point being proportional to the degree of the corresponding representation. Basic in our work is a certain countably-additive non-negative function, which we call the "dual gage" of G, defined on the lattice of all projections in the weak closure V of the algebra generated by left translations in L2(G),-or alternatively, on the lattice of closed linear manifolds in L2(G) invariant under right translations. The dual gage, which is defined in an intrinsic fashion, is invariant under transformation by unitary operators in V, and is in fact a "weight function" in the terminology of von Neumann [6]. His reduction theory consequently yields a representation of the dual gage as an integral over the measure space described above of constituents (relative dimension functions) arising from factors. This representation together with the facts that the convolution operator on L2(G) defined by a self-adjoint element of L2(G) is hypermaximal symmetric (Ambrose [1]) and that every bounded linear operator on L2(G) which commutes with all right translations is in V (Segal [4]), are the principal known results used in our derivation of the generalized Plancherel formula.