The primary purpose of this
paper is to investigate positive functionals on and representations of complete locally
m-convex algebras with a continuous involution with emphasis on the special case of
commutative algebras.
The first part is a study of the continuous positive functionals on a complete
locally m-convex (LMC) algebra A with identity and continuous involution. It is
shown that the LMC equivalent of “positive face of the unit ball” in A∗ is a
w∗-closed convex set and is the closed convex hull of its extreme points, which
are the normalized indecomposable continuous positive functionals on A.
This is applied to the commutative case to obtain a representative of these
functionals as integrals on the space Φ∗ of symmetric maximal ideals of
A.
The second part is an investigation of representations of an LMC algebra A in
B(H). Necessary and sufficient conditions in order that a cyclic representation be
continuous are given. For normed algebras completeness guarantees the continuity of
all representations. An example shows that this is not the case for LMC algebras. It
is shown that cyclic representations of commutative algebras are equivalent to left
multiplication on a suitably chosen L2-space over Φ∗, and that the operators can be
represented as norm-convergent integrals with respect to a compactly-supported
spectral measure on Φ∗. These results are then extended to general continuous
representations of LMC algebras.
