CO-UNIVERSAL C*-ALGEBRAS ASSOCIATED TO APERIODIC k-GRAPHS. (September 2014)

• Record Type: Journal Article
• Authors: KANG, SOORAN; SIMS, AIDAN
• Is Part Of: Glasgow mathematical journal. Volume 56:Part 3(2014:Sep.)
• Page Start: 537
• Page End: 550
• Publication Date: 2014-09
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1017/S001708951300044X
• Languages: English
• ISSNs: 0017-0895

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>We construct a representation of each finitely aligned aperiodic <italic>k</italic>-graph Λ on the Hilbert space <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239rv2b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{H}^{\rm ap}$]]></tex-math></alternatives></inline-formula> with basis indexed by aperiodic boundary paths in Λ. We show that the canonical expectation on <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1239rtz5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\mathcal{B}(\mathcal{H}^{\rm ap})$]]></tex-math></alternatives></inline-formula> restricts to an expectation of the image of this representation onto the subalgebra spanned by the final projections of the generating partial isometries. We then show that every quotient of the Toeplitz algebra of the <italic>k</italic>-graph admits an expectation compatible with this one. Using this, we prove that the image of our representation, which is canonically isomorphic to the Cuntz–Krieger algebra, is co-universal for Toeplitz–Cuntz–Krieger families consisting of non-zero partial isometries.</p> </abstract>