A GELFAND-NAIMARK THEOREM FOR SOME C-ALGEBRAS WITH FINITE DIMENSIONAL IRREDUCIBLE REPRESENTATIONS

One of the most basic and elegant results of the theory of C-algebras is
the commutative Gelfand-Naimark theorem. Among the earliest studies that endeavoured
to generalize it is the seminal paper of Fell [8] where the C-algebras
are represented as algebras of operator fields. This method was especially successful
in representing the n-homogeneous C-algebras as algebras of continuous
cross-sections of vector bundles whose fiber is Mn, the algebra of all complex
n × n matrices, and whose group is the group of all the automorphisms of Mn,
a result that was obtained independently in [12]. The program was continued in
[5], [11] and [7] and many other papers. A different approach was taken in [10].
Here we present a generalization of the result on n-homogeneous C-
algebras mentioned above to C-algebras that have a Hausdorff spectrum and
only finite dimensional irreducible representations. To this end we define and
investigate in Section 2 a kind of Banach bundle whose fiber is a variable normed
linear space of finite dimension. Section 3 contains the main representation result.
The fibers are specialized to full algebras of complex matrices and the base
spaces are required to be locally compact Hausdorff spaces. It is shown that for
418 ALDO J. LAZAR
the kind of Banach bundles we discuss in this section, the algebras of all continuous
cross-sections that vanish at infinity have only finite dimensional irreducible
representations that are given by evaluations at the points of the base
space. Moreover, every C-algebra that has only finite dimensional irreducible
representations and a Hausdorff spectrum is isomorphic to such an algebra of
continuous cross-sections. The section ends with a condition for the isomorphism
of the algebras of continuous cross-sections considered in this paper. In Section 4
we produce a formula for the upper and lower multiplicities of an irreducible representation
that were defined in [2] and [4]. One of the reasons that makes these
invariants interesting is that they measure the extent to which Fell’s condition can
fail, see Theorem 4.6 of [2]. As shown in Proposition 10.5.8 of [6], for C-algebras
with Hausdorff spectrum the presence of Fell’s condition is necessary and suffi-
cient for having continuous trace. Another result in the same section is a representation
for the multiplier algebra of an algebra of continuous cross-sections of
the kind studied here.