George W. Mackey

Induced Representations of Locally Compact Groups II. The Frobenius Reciprocity Theorem

In the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by Frobeniuss notion of induced representation. To every representation L of the subgroup G of the finite group 65 there is assigned a well defined representation UL of ( called the representation of M induced by L. In I of this series [10] the author began a systematic study of a generalization of this notion in which (5 is a separable locally compact topological group and the spaces of L and UL are possibly infinite dimensional Hilbert spaces. In particular [10] contains a generalization of the Frobenius reciprocity theorem; that is the theorem asserting that UL contains the irreducible representation M of (D just as many times as the restriction of M to G contains the irreducible representation L of G. The generalization contained in [10] is unsatisfactory in that it deals only with the discrete finite dimensional irreducible components of the representations concerned and becomes vacuous when these representations decompose in a continuous fashion or have no finite dimensional irreducible components. A more satisfactory generalization has been obtained by Mautner [13, 14]. It deals in a consequent fashion with continuously decomposable representations of 5 but is restricted by the requirement that G be compact. This means in particular that only discretely decomposable representations of G need be dealt with. The principal result of the present article is a generalization of the Frobenius reciprocity theorem which deals effectively with continuously decomposable representations of both M and G. Since in compensation for the hypothesis that G be compact, we require that both G and 5 have regular representations which are of type I, our theorem does not quite include that of Mautner. However we show in addition that whenever the regular representation of G is not only of type I but also discretely decomposable then the requirement that the regular representation of (M be of type I can be eliminated. Thus our methods also yield a result which includes that of Mautner. In neither of our results is it necessary to assume that the groups concerned are unimodular. A noteworthy feature of our principal theorem is that it includes a reciprocity not only for the multiplicities but for the measures involved in the continuous decompositions as well. The basic idea in our approach is a new proof of the Frobenius reciprocity theorem in the finite case which has the advantage of generalizing significantly

Harmonic analysis as the exploitation of symmetry–a historical survey

Preface

Ring and lattice characterizations of complex Hilbert space

SHIZUO KAKUTANI AND GEORGE W. MACKEY Introduction. In an earlier paper [l] by the authors it was suggested that at least the ring characterization of real Hubert space given therein might be extended to the complex case by making use of a device employed by B. H. Arnold [2] in so extending a theorem of Eidelheit. It is the purpose of the present note to show that this can indeed be done not only for the ring characterization but for the lattice one as well. The difficulty in the complex case is that the complex field admits a great many discontinuous automorphisms. It is overcome by making use of the device of Arnold mentioned above to show that in the infinite-dimensional case only continuous automorphisms present themselves (see Lemma 2 below). It is shown by an example that the infinite-dimensionality is essential and that accordingly the theorems of [l] cannot be extended to the complex case in quite their full generality.

Equivalence of a problem in measure theory to a problem in the theory of Vector Lattices

In connection with some other work of the author a question arises concerning the form of the general bounded linear functional on certain vector lattices. It is the purpose of this note to show that this question is completely equivalent to a question in measure theory which has been discussed and partially answered by Ulam [2]. Let 5 be an abstract set and let % be the vector lattice of all realvalued functions defined on S. For each So in S the function F on gf such that F(f) =/(so) for all ƒ in g? is clearly a linear functional. We shall call it the point functional belonging to s0 or simply a point functional. Obviously every point functional and hence every finite linear combination of point functionals is bounded in the sense that it carries every bounded subset of g into a bounded set of real numbers. Our question is as to whether every bounded linear functional on § is a finite linear combination of point functionals. We shall show that this is the case if and only if there exists no countably additive measure a which is defined for all subsets of 5, which is zero at points, which takes on only the values zero and one, and which does not vanish identically. It is well known that every bounded linear functional on a vector lattice is a difference of non-negative linear functionals and it is obvious that a non-negative linear functional is bounded. It follows that we need only consider non-negative linear functionals. Passages from a measure to a non-negative linear functional defined on a class of functions and vice versa are of frequent occurrence in mathematical literature. The proof of our theorem rests basically on the fact that when the methods used in effecting these passages are applied to the case at hand one obtains a natural one-to-one correspondence between the non-negative linear functionals on g and the countably ad-

The Mathematical Papers

Eugene Wigner is above all a theoretical physicist. However he was one of the two men (Hermann Weyl was the other) who introduced a powerful new mathematical tool into quantum mechanics in its earliest years. This is the theory of group representations, invented by Frobenius in 1896, and apparently not applied outside of pure group theory until E. Artin’s startling application to number theory in 1923. Wigner’s first application of this theory to quantum mechanics was published only four years later in 1927. Weyl’s contribution was of a completely different character and was made a few months after Wigner’s.

Induced Representations of Locally Compact Groups and Applications

Roughly the first half of this paper is an introductory exposition of some of the main ideas of the theory of induced representations with special emphasis on the influence of the work of M. H. Stone. The second half deals with applications and with certain extensions and refinements of known results demanded by these applications.

Functions on locally compact groups

1. Background. The subject of this address is a branch of mathematics which may be regarded as a combination of the classical theory of representations of finite groups by matrices and that part of analysis centering around the theory of Fourier series and integrals. The connection between these two apparently diverse subjects arises simply enough from the fact that the real numbers and the real numbers modulo 27T form groups under addition. In one form of the theory of representations of a finite group G a central role is played by the so-called group ring or group algebra. This is usually defined as the set of all formal linear combinations of group elements C1S1+C2S2+ • • • +cnsn, where each S;£G and each d is a complex number. Two such expressions are added in the obvious manner and are multiplied by writing down the formal product and simplifying by means of the distributive law and the given multiplication of group elements. I t may also be defined (and this is the definition we shall use) as the vector space of all complex-valued functions on G with multiplication defined by the formula

Weyl’s Program and Modern Physics

The author is presently engaged in trying to fit as much as possible of modern elementary particle physics into a group theoretical framework described at some length in the reference cited at the end of section IV. This article is in part an exposition of the relationship of this program to one formulated by Weyl in 1927, in part a review and revision of material from the reference mentioned above and in part a description of ideas and incomplete work relevant to carrying the program further.

A remark on locally compact Abelian groups

I t has recently been shown by Halmos [l ] that there exists a compact topological group which is algebraically isomorphic to the additive group of the real line, an example being given by the character group of the discrete additive group of the rationals. Exploiting his argument a bit further it is easy to see that the most general such example is the direct sum of N replicas of the one already given where ^ is a cardinal such that 2^ ^ C. This having been observed it naturally occurs to one to ask for the most general locally compact topological group with the algebraic structure in question. I t is the purpose of the present note to give a complete answer to this question. We shall do so by giving a proof of the following theorem.

Infinite Dimensional Group Representations and their Applications

The theory of infinite dimensional group representations may be regarded as a blend of Fourier analysis and the classical purely algebraic theory of group representations.