Document Type: Research Paper

Authors

1 Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, 15914 Tehran, Iran.

2 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Iran.

Abstract

In this paper, we describe the primitive ideal space of the $C^*$-algebra $C^*(\mathcal G)$  associated to the ultragraph $\mathcal{G}$. We investigate the structure of the closed ideals of the quotient ultragraph $C^*$-algebra  $C^*\left(\mathcal G/(H,S)\right)$ which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the Hong and Szyma$\acute{ \mathrm { n } }$ski's description of the primitive ideal space of a graph $C ^ *$-algebra by a simpler method.

Keywords

Main Subjects

###### ##### References

[1] G. Abrams, P. Ara, and M. Siles Molina, Leavitt Path Algebras, Lecture Notes in Mathematics Vol. 2191, Springer, London, 2017.

[2] T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The $C^*$-algebras of row-finite graphs, New York J. Math., 6 (2000), pp. 307-324.

[3] T.M. Carlsen, S. Kang, J. Shotwell, and A. Sims, The primitive ideals of the Cuntz-Krieger algebra of a row-finite higher-rank graph with no sources, J. Funct. Anal., 266 (2014), pp. 2570-2589.

[4] T.M. Carlsen and A. Sims, On Hong and Szymanski's description of the primitive-ideal space of a graph algebra, Operator algebras and applicationsthe Abel Symposium (2015), Abel Symp., 12, Springer, [Cham], (2017), pp. 115-132.

[5] J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math., 56 (1980), pp. 251-268.

[6] R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math., 512 (1999), pp. 119-172.

[7] N. Fowler, M. Laca, and I. Raeburn, The $C^*$-algebras of infinite graphs, Proc. Amer. Math. Soc., 128 (2000), pp. 2319-2327.

[8] J. Hong and W. Szymanski, The primitive ideal space of the $C^*$-algebras of infinite graphs, J. Math. Soc. Japan, 56 (2004), pp. 45-64.

[9] T. Katsura, P.S. Muhly, A. Sims, and M. Tomforde, Utragraph $C^*$-algebras via topological quivers, Studia Math., 187 (2008), pp. 137-155.

[10] A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math., 184 (1998), pp. 161-174.

[11] H. Larki, Primitive ideals and pure infiniteness of ultragraph $C^*$-algebras, J. Korean Math. Soc., 56 (2019), pp. 1-23.

[12] H. Larki, Primitive ideal space of higher-rank graph $C^*$-algebras and decomposability, J. Math. Anal. Appl., 469 (2019), pp. 76-94.

[13] M. Tomforde, A unified approach to Exel-Laca algebras and $C^*$-algebras associated to graphs, J. Operator Theory, 50 (2003), pp. 345-368.