Modulars from Nakano onwards

Abstract

We discuss and compare a number of notions of modulars appeared in literature, among which there is a selection of the well known ones. We highlight the connections between the various definitions and provide several examples, taken from existing literature, recalling known results and completing the picture with some original considerations.

Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

“…by assumption (see also [10]). Since H 0 (K ) is dense in H (K ), thus defined U(g) can be uniquely extended to H (K ).…”

“…can be demonstrated to be a well-defined inner product (more information on the correctness of such definitions can be found e.g. in [10]). Then one shows that the completion of H 0 (K ) in the norm induced by that inner product, denoted H (K ) := H 0 (K ) = span{K a } a∈A , can still be interpreted as a space of complex-valued functions on A, with K as its reproducing kernel.…”

“…Many authors [8,18,22] use a more general and somewhat more involved definition of a unitary representation of a groupoid, in which H is a Hilbert bundle or a measurable field of complex Hilbert spaces. This allows for constructing many more interesting Hilbert spaces from the fibers H x than just the completed direct sum H above (see also [10]). In the present paper, however, we keep the definition simple so that the constructions of the positive definite kernels and RKHS become more tractable.…”

“…where the maps K γ : → C and K x γ : x → C have been identified (the former being the extension by zero of the latter). Since both above spaces are equipped with the same inner product (given by (2)), they yield the same Hilbert spaces after completion (see also [10]).…”

“…Example 10 This example is inspired by [4]. Let X be the set of all pairs ( , z), where is a domain (nonempty, open and connected set) in C n and z ∈ .…”