nLab homogeneous C-star-algebra

Idea

A C*-algebra is called $n$-homogeneous (or n-dimensionally homogeneous) if each irreducible $C^\star$-representation of the algebra is $n$-dimensional.

Related entries

• Azumaya algebra
• Fell bundle

References

• Jun Tomiyama, Masamichi Takesaki, Applications of fibre bundles to the certain class of $C^\ast$-algebras, Tohoku Math. J. (2) 13:3 (1961) 498–522 euclid doi

Let $A$ be a C${}^\ast$-algebra. If the set of pure states of $A$ is compact and that of primitive ideal?s which are the kernels of one-dimensional irreducible reprentations forms an open set in the structure space of $A$, then $A$ is isomorphic to the C${}^\ast$-sum of a finite number of homogeneous C${}^\ast$-algebras.

• Shaun Disney, Iain Raeburn, Homogeneous C${}^\ast$-algebras whose spectra are tori, J. Australian Math. Soc. 38:1 (1985) 9–39 doi

The comparison of the Artin’s theorem on characterization of Azumaya algebras and Tomiyama-Takesaki’s theorem on $n$-homogeneous $C^\ast$-algebras is in chapter 9 of

• Edward Formanek, Noncommutative invariant theory, in: Group actions on rings (Brunswick, Maine, 1984), 87–119, Contemp. Math. 43, Amer. Math. Soc. 1985 doi

category: operator algebras