Ray A. Kunze

Bessel functions and representation theory. I

Abstract In this paper a general theory of operator-valued Bessel functions is presented. These functions arise naturally in representation theory in the context of metaplectic representations, discrete series, and limits of discrete series for certain semi-simple Lie groups. In general, Bessel functions J λ are associated to the action by automorphisms of a compact group U on a locally compact abelian group X , and are indexed by the irreducible representations λ of U that appear in the primary decomposition of the regular representation of U on L 2 ( X ). Then on the λ-primary constituent of L 2 ( X ), the Fourier transform is described by the Hankel transform corresponding to J λ . More detailed information is available in the case in which ( U , X ) is an orthogonal transformation group which possesses a system of polar coordinates. In particular, when X = F k × n , F a real finite-dimensional division algebra, with k ⩾ 2 n and O ( k , F ), the representations λ of U are induced in a certain sense from representations π of GL( n , F ). This leads to a characterization of J λ as a reduced Bessel function defined on the component of 1 in GL( n , F ) and to the connection between metaplectic representations and holomorphic discrete series for the group of biholomorphic automorphisms of the Siegel upper half-plane in the complexification of F n × n .

Bessel functions and representation theory, II holomorphic discrete series and metaplectic representations

Abstract This paper presents representation-theoretic applications of the general theory of operator-valued Bessel functions developed in the first paper of this series. Here, the concern is with the circle of ideas relating decomposition of the Fourier transform on F k × n, F a real finite-dimensional division algebra and k ⩾ 2n, to metaplectic representations, holomorphic discrete series, and limits of holomorphic discrete series for the group of biholomorphic automorphisms of the Siegel upper half-plane in the complexification of F n × n.

C*-Algebras and Applications

The theory of C*-algebras dates from the discovery by Gelfand and Naimark that uniformly closed self-adjoint operator algebras on Hilbert space—unlike the rings studied by von Neumann and Murray—could be characterized in simple intrinsic algebraic terms, independently of their action on Hilbert space. This opened up the study of the algebraic isomorphism classes of such algebras, in the sense of emphasizing its cogency. It was soon found that C*-algebras have certain applications in quantum mechanics, and especially in quantum field theory, in parts of group representation theory, and some other areas, in which W*-algebras could not be substituted.

A note on square-integrable representations

Abstract In this paper we prove that every square-integrable representation of a Hilbert algebra is a direct sum of irreducible such representations, and we note that this implies that every square-integrable representation of a locally compact unimodular group is a direct sum of irreducible square-integrable representations.

Convergence and Differentiation

The reader is probably familiar with the extensive and penetrating theory of finite-dimensional linear spaces and operators thereon. One cannot hope to obtain a similarly penetrating theory when the vector space is infinite-dimensional without additional structure in the space. A first step in civilizing a linear space for the purposes of analysis is the introduction of a convenient topology, i.e. one with respect to which relevant operations are continuous. A natural candidate for such a topology in the linear space M of all complex-valued locally measurable functions is that of sequential pointwise convergence. However, this topology may fail to have cogent structural properties. One is led rather to deal with a modification \(\tilde M\) of M which is obtained by identifying functions which agree except on local null sets; and which is given the topology it inherits naturally from that of pointwise sequential convergence in M. It will be seen in fact that \(\tilde M\) is metrizable in this topology, in the key case of a finite measure space; and that it becomes a topological linear space in the following sense.

The Trace as a Non-Commutative Integral

The philosophy of the theory of integration has evolved considerably in the past several decades. Newer applications have tended to de-emphasize point-wise features in favor of global and/or algebraic ones. This has led in particular to a realization that the theory of the trace on the ring of all bounded operators on a Hilbert space was parallel in a number of ways to abstract Lebesgue integration theory, and to the development of a natural simultaneous generalization of both theories. In these sections we first develop the theory of the trace in B(H) from a standpoint that emphasizes the integration-theoretic analogy on the one hand and points towards the generalization to arbitrary rings on the other. We should point out however that the basic results concerning the trace on B(H) were originally obtained quite independently of these considerations, because of their intrinsic interest, and that these results are important for a variety of applications.

Operator Rings and Spectral Multiplicity

Operator algebra in Hilbert space has developed extensively from a variety of motivations and in several directions, beginning with the double-commutor theorem of von Neumann (1930). Von Neumann himself, in part in collaboration with F. J. Murray, founded during the ensuing two decades the theory of what he called simply “rings”, but which are known more specifically as “ W*-algebras” or “von Neumann algebras”. Here we shall follow von Neumann’s usage of the term “ring” when there appears to be no likelihood of confusion, and otherwise employ the term “ W*-algebra”, the “ W” referring to the weak topology, in which the algebra is closed, and the “*” to its self-adjointness.

Spectral Analysis in Hilbert Space

From the last result of the preceding chapter, a good deal of nontrivial information can be obtained about self-adjoint operators on Hilbert space, incomparably more than is available for operators in Banach spaces, other than those of special compactness properties, or for arbitrary (non-self-adjoint) operators on Hilbert space. This result fails, however, to give a structure theorem for a self-adjoint operator on a Hilbert space, i.e., a result giving a simple explicit form for the operator, within the isomorphism appropriate to Hilbert space (unitary equivalence). What one would like, of course, is an analog in Hilbert space to the diagonalization of a self-adjoint operator in a finite-dimensional unitary space. Simple examples such as the operation of multiplication by x, acting on L2(0,1), show that a direct analog does not exist: there need be no eigenvalues or eigenspaces whatever. However, this example points the way to an appropriate analog, which is that in which invariant subspaces are decomposed, not into a discrete direct sum of eigenspaces, but into a continuous direct integral of infinitesimal eigenspaces, any one of which corresponds to a point of measure zero in the formation of the direct integral.

Measurable Functions and Their Integrals

For a given basic measure space, the basic measurable sets and the corresponding step functions form quite limited classes. Many of the sets, and actually most of the functions, which arise in analytical theory and practice will not be in these classes. It is apparent, for example, in the case of the Lebesgue basic measure space on the real line, that the integration of bounded continuous functions over finite intervals is not covered by the theory of integration for step functions. This is, however, as it should be. The general program in the development of the theory of integration should commence with a limited, yet transparent and readily constructed notion of integration such as that given in Chap. 2. The next step is to extend the integral to a wide class of functions, including, hopefully, all those of analytical interest, in such a way as to maintain all such useful properties as those already derived. That this should be possible is not at all obvious, nor does it have even an appearance of intuitive inevitability. That it is possible in fact, in such a relatively unique fashion, is in its way a quite striking phenomenon.

Locally Compact and Euclidean Spaces

Except in the examples, the set S on which the measures have been defined (more precisely, on certain subsets of which it has been defined) has been an abstract set, devoid of any special structure. In the particular case in which S has additionally the structure of a topological space, e.g., when S is euclidean space, it is natural to consider the relations between the topological and measure-theoretic features of S. For example, elementary analysis suggests that continuous functions should be measurable; this depends, however, on the existence of a suitable relationship between the measure and the topology.