Conjugacy of circle actions on UHF C*-algebras

Question

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?

Answer 1

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.