Conjugacy of circle actions on UHF C*-algebras Consider pointwise continuous actions of the unit circle on the $2^{\infty}$-UHF C*-algebra A by *-automorphisms. Assume that two such actions have the same fixed point algebra, i.e., elements that are fixed elementwise by the whole circle action. Are these actions conjugate in some sense? What if the fixed point algebra is the canonical masa of A?
 A: The question is a bit vague, but I will give uncountably many
non-conjugate actions of the circle group on the $2^{\infty}$ UHF
C*-algebra whose fixed point algebra is a standard masa (corresponding to
the product type decomposition).
The first class will be product type actions, where we allow tensor
products with arbitrary choices of matrix sizes, subject to being powers
of $2$.
Let $(p_n)$ be a sequence of polynomials with nonnegative coefficients,
such that in any product of finitely many of them, each monomial is
uniquely expressible as a product of the monomials in the individual
$p_n$s; we call such a sequence non-interacting. For example, $p_n = 1 +
x^{2^{n}}$, but there are lots of such.
Non-interacting polynomials play a major role in classification of ergodic ($\bf Z$-)
actions up to measure-theoretic equivalence, but here we are involved with
the topological equivalence. So we also insist on the following.
In addition to $(p_n)$ be non-interacting, we require all nonzero
coefficients to be $1$, and for each $n$, $p_n(1)$ is a power of $2$ (the
powers can vary completely arbitrarily). Of course, $p_n(1)$ counts the
number of monomials appearing in $p_n$.
Each $p_n$ is thus a character of the circle, ${\bf T}$, and is the character
of the diagonal representation $\pi_n: z \mapsto {\rm diag} (z^{m(i,n)})$,
where the $m(i,n)$ run through all the exponents in $p_n$. The fact that
$p_n (1)$ is a power of $2$  means that the representation has degree a
power of $2$, so we can take the infinite tensor product, $\alpha:=
\otimes \text{Ad}\,\pi_n$ acting on the infinite tensor product of the
matrix algebras; as the latter all have size a power of two, this gives an
action, $\alpha$, of the circle group on the $2^{\infty}$ UHF.
Now any finite tensor product of the $\pi_n$ has distinct exponents
(because of the unique expression, every coefficient of the product of the
corresponding $p_n$ is either zero or $1$). Hence the centralizer of the
image of $z$ is exactly all the diagonal matrices in that tensor product.
Taking the limit, we see that the centralizer is exactly the standard masa
(diagonal matrices) for the particular tensor product realization of the
UHF.
The classification of this type of action (and much more general ones)
is given in [HR] Handelman and Rossmann, Actions of compact groups on AF
$C^\ast$-algebras,  Illinois J Math  29  (1985),  no. 1, 51–95. The
complete invariant, yielding conjugacy, is given by the ordered K$_0$
group of the  crossed product as an ordered  ${\bf Z}[x,x^{-1}]$-module (of
course, ${\bf Z}[x,x^{-1}]$ is just the representation ring of the circle; the construction and invariants extend to all compact groups),
together with a fixed order unit, and outer conjugacy and various others
are given by removing various stuff.
Moreover, the ordered group of the crossed product is just the direct
limit of multiplications, $\times p_n :{\bf Z}[x,x^{-1}] \to {\bf Z}[x,x^{-1}]$.
Forgetting the ordering, merely as a module, this is rank one, and it is
fairly easy to construct uncountably many different module structures (let
along ordered modules) just using the sequences $(p_n)$. [This is
analogous to, but somewhat more complicated than, the supernatural number
that classifies rank one noncyclic subgroups of the rationals.]
All product type actions whose fixed point subalgebra is a standard masa
(there are nonstandard masas, which should be avoided) arise in this
fashion. However, there must also be non-product actions on the UHF,
arising from actions of the tori on direct limit of nonsimple things
(whose union closed is the UHF), and then the module rank can be greater
than one. [But even if the module rank is one, it does not imply the
action is conjugate to a product type; the ordering plays an essential
role; see my paper, Imitation product-type actions on UHF algebras,  J
Algebra  99  (1986),  no. 1, 1–21.]
But there is still more. I don't know whether they exist (they probably
do): actions of the circle with masas as fixed point algebras that are not
even locally representable (meaning that they do not act on any increasing
union of finite dimensional subalgebras ...; see [HR] for the actual
definition---this answer has gone on too long.)
Anyway, the upshot is that it is the crossed product, not the fixed point
algebra, which when its K$_0$ group is viewed as an ordered module,
contains almost all the  information for classification.
