Bimodules

We use the technique of bimodules analogous to the technique by Paul Muhly and Baruch Solel.

. But: by the formulas:

An ideal I in A is a -bimodule in the following sense:

Idea: Consider all closed bimodules like above -- not only ideals.

Warning: G is not r-discrete! Old methods don't work. Not all elements are functions on G etc.

(In r-discrete care all elements are functions on G. Example: A = matrices . Then .)

Idea: An element of A may not be a function on G like in the r-discrete case -- but it has its support: the complement of its zero set.

A bimodule has its support -- the ``closure'' of the union of the supports of all its elements.

Consider the following passage:

where

IDEA OF THE PROOF.

-- from the definition.
-- difficult.

1. Consider with -- the translation. The algebra arising is a subalgebra of .

   Observation: The algebra norm is dominated by the -norm on .

   We take and prove that f can be approximated by functions from B.

   Consider only good bimodules -- such that:
   (1) ;
   (2) is the union of graphs.

   Consider the following cases.

   1. S is a rectangle.

      -- a function. But: .
      ``Cut down'' g to the rectangle S by multiplications on right and on left.

      g can be approximated in -norm by functions from B. Using Case 1.

   2. S is of general type.

      -- unions of rectangles, and any can be approximated by . Using (2) and the fact that f is compact.
      Not trivial at all!!!

2. Now consider general dynamical systems.

   Use the theorem by Gootman and Rosenberg [1, Theorem 1.4,]: locally the orbits are ``parallel''. Therefore G has simple local structure.

   Again, consider two subcases.

   1. The semidynamical system is a subsystem of a ``full'' dynamical system. Then is an equivalence relation.

      All the definitions (support, good bimodule etc.) make sense orbitwise. The norm is also defined orbitwise.

      The spectral theorem for bimodules can be reduced to a single orbit, and then to the case `` on '' considered before.
      Again, not trivial!

   2. General semidynamical system.

      Dilate to a full dynamical system. Every map must be surjective.

      -- new system. Elements -- ``backward semiorbits'' of . Every can generate many .

      Action :
      for t > 0 as ;
      for t < 0 by ``cutting the backward semiorbit''.

      The spectral theorem is proved by reducing to previous case.