## A suspension theorem for continuous trace $C^ *$-algebras

HTML articles powered by AMS MathViewer

- by Marius Dădărlat
- Proc. Amer. Math. Soc.
**120**(1994), 761-769 - DOI: https://doi.org/10.1090/S0002-9939-1994-1166354-3
- PDF | Request permission

## Abstract:

Let $\mathcal {B}$ be a stable continuous trace ${C^{\ast }}$-algebra with spectrum $Y$. We prove that the natural suspension map ${S_{\ast }}:[{C_0}(X),\mathcal {B}] \to [{C_0}(X) \otimes {C_0}({\mathbf {R}}),\mathcal {B} \otimes {C_0}({\mathbf {R}})]$ is a bijection, provided that both $X$ and $Y$ are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and $X$ is noncompact. This is used to compute the second homotopy group of $\mathcal {B}$ in terms of $K$-theory. That is, $[{C_0}({{\mathbf {R}}^2}),\mathcal {B}] = {K_0}({\mathcal {B}_0})$, where ${\mathcal {B}_0}$ is a maximal ideal of $\mathcal {B}$ if $Y$ is compact, and ${\mathcal {B}_0} = \mathcal {B}$ if $Y$ is noncompact.## References

- Jacques Dixmier,
*$C^*$-algebras*, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR**0458185** - M. Dădărlat and A. Némethi,
*Shape theory and (connective) $K$-theory*, J. Operator Theory**23**(1990), no. 2, 207–291. MR**1066807** - Edward G. Effros,
*On the structure theory of $C^{\ast }$-algebras: some old and new problems*, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 19–34. MR**679693** - Dale Husemoller,
*Fibre bundles*, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR**0229247**, DOI 10.1007/978-1-4757-4008-0 - Jonathan Rosenberg,
*The role of $K$-theory in noncommutative algebraic topology*, Operator algebras and $K$-theory (San Francisco, Calif., 1981) Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 155–182. MR**658514** - Claude Schochet,
*Topological methods for $C^{\ast }$-algebras. III. Axiomatic homology*, Pacific J. Math.**114**(1984), no. 2, 399–445. MR**757510**, DOI 10.2140/pjm.1984.114.399 - Graeme Segal,
*$K$-homology theory and algebraic $K$-theory*, $K$-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975) Lecture Notes in Math., Vol. 575, Springer, Berlin, 1977, pp. 113–127. MR**0515311** - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR**0210112**

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**120**(1994), 761-769 - MSC: Primary 46L85; Secondary 19K99, 46L80, 55P99
- DOI: https://doi.org/10.1090/S0002-9939-1994-1166354-3
- MathSciNet review: 1166354