## Annals of Functional Analysis

### Scattered locally $C^{\ast}$-algebras

Maria Joiţa

#### Abstract

In this article, we introduce the notion of a scattered locally $C^{\ast}$-algebra and we give the conditions for a locally $C^{\ast}$-algebra to be scattered. Given an action $\alpha$ of a locally compact group $G$ on a scattered locally $C^{\ast}$-algebra $A[\tau_{\Gamma}]$, it is natural to ask under what conditions the crossed product $A[\tau_{\Gamma}]\times_{\alpha}G$ is also scattered. We obtain some results concerning this question.

#### Article information

Source
Ann. Funct. Anal. Volume 9, Number 1 (2018), 30-40.

Dates
Received: 12 October 2016
Accepted: 30 January 2017
First available in Project Euclid: 12 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1499824814

Digital Object Identifier
doi:10.1215/20088752-2017-0021

#### Citation

Joiţa, Maria. Scattered locally $C^{\ast}$ -algebras. Ann. Funct. Anal. 9 (2018), no. 1, 30--40. doi:10.1215/20088752-2017-0021. https://projecteuclid.org/euclid.afa/1499824814

#### References

• [1] B. Blackadar, Operator Algebras: Theory of $C^{\ast}$-Algebras and von Neumann Algebras, Encyclopaedia Math. Sci. 122, Springer, Berlin, 2006.
• [2] C. H. Chu, Crossed products of scattered $C^{\ast}$-algebras, J. Lond. Math. Soc. (2) 26 (1982), no. 2, 317–324.
• [3] M. Fragoulopoulou, Topological Algebras with Involution, North-Holland Math. Stud. 200, North-Holland, Amsterdam, 2005.
• [4] M. Haralampidou, “The Krull nature of locally $C^{\ast}$-algebras” in Function Spaces (Edwardsville, Ill., 2002), Contemp. Math. 328, Amer. Math. Soc., Providence, 2003, 195–200.
• [5] T. Huruya, A spectral characterization of a class of $C^{\ast}$-algebras, Sci. Rep. Niigata Univ. Ser. A 15 (1978), 21–24.
• [6] A. Inoue, Locally $C^{\ast}$-algebra, Mem. Fac. Sci. Kyushu Univ. Ser. A, 25 (1971), 197–235.
• [7] H. E. Jensen, Scattered $C^{\ast}$-algebras, Math. Scand. 41 (1977), no. 2, 308–314.
• [8] H. E. Jensen, Scattered $C^{\ast}$-algebras, II, Math. Scand. 43 (1978), no. 2, 308–310.
• [9] M. Joiţa, Crossed Products of Locally $C^{\ast}$-Algebras, Editura Academiei Române, Bucharest, 2007.
• [10] M. Kusuda, A characterization of scattered $C^{\ast}$-algebras and its application to $C^{\ast}$-crossed products, J. Operator Theory 63 (2010), no. 2, 417–424.
• [11] M. Kusuda, $C^{\ast}$-algebras in which every $C^{\ast}$-subalgebra is AF, Quart. J. Math. 63 (2012), no. 3, 675–680.
• [12] A. J. Lazar, On scattered $C^{\ast}$-algebras, in preparation.
• [13] A. Pelczynski and Z. Semadeni, Spaces of continuous functions, III: Spaces $C(\Omega)$ for $\Omega$ without perfect subsets, Studia Math. 18 (1959), 211–222.
• [14] N. C. Phillips, Inverse limits of $C^{\ast}$-algebras, J. Operator Theory 19 (1988), no. 1, 159–195.
• [15] M. L. Rothwell, Scattered $C^{\ast}$-algebras, in preparation.
• [16] W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39–42.