The temperature state space of a $C^\ast$-dynamical system. II


The space of KMS states at finite inverse temperatures of a dynamical system $(A,\gamma)$, where $A$ is a simple uital $C^\ast$-algebra and $\gamma$ a faithful representation of a Lie group $\Gamma$ as automorphisms of $A$, is a locally compact bundle of simplices over the space of finite inverse temperatures, which may be identified with the Lie algebra of $\Gamma$, such that the space of continuous real-valued functions on this bundle which are affine on fibres separates points. In the case that the Lie algebra of $\Gamma$ is $\mathbf{R}$, we shows in [4] that any metrizable such bundle of simplices, in this case over $\mathbf{R}$, arises in this way. Here we extend this to the case $\mathbf{R}^n$. Roughly speaking, this demonstrates the occurrence of arbitrary phase diagrams with finitely many parameters, which could be thought of as thermodynamical variables such as temperature and chemical potential.

When one considers also infinite inverse temperatures and the corresponding ground states, one obtains a compactification fo the above bundle, the new fibres of which lie over the sphere at infinity, and are convex sets but not necessarily simplices.

Extending a result for ground states in [4] in the case that the Lie algebra of $\Gamma$ is $\mathbf{R}$ we show that when the Lie algebra of $\Gamma$ is $\mathbf{R}^n$, any totally disconnected closed subset $T_\infty$ of the sphere at infinity may arise as the set of infinite inverse temperatures at which ground states exist, and the corresponding ground state spaces may be mutually disjoint and isomorphic as a bundle of compact convex sets over $T_\infty$ to the bundle of state spaces of simple quotients of an arbitrary unital separable approximately finite-dimensional $C^\ast$-algebra with primitive spectrum $T_\infty$.

The step from one parameter to two (or more) requires new techniques.


Ola Bratteli

George A. Elliott

Akitaka Kishimoto