The uniqueness of Cuntz-Krieger type algebras

On certain properties of Cuntz–Krieger-type algebras

“…We briefly recall the basic definitions and facts of the class of Cuntz-Krieger type algebras introduced in [4] and slightly extended in [6].…”

“…(A) There exists a gauge action t : H −→ Aut(F/I) determined by t λ (a) = λ a a for all a ∈ A and λ = (λ b ) b∈A ∈ H. Denote by (Ĥ, +, 0) the character group of (H, ·, 1); note that we write the group operation ofĤ additively. The gauge action t induces a so-called balance function bal : W \{0} −→Ĥ from the nonzero words of F/I to the character groupĤ determined by bal(a)((λ b ) b∈A ) = λ a ∈ T, bal(xy) = bal(x) + bal(y) and bal(x * ) = −bal(x), where a ∈ A, (λ b ) b∈A ∈ H ⊆ T A and x, y ∈ W (see [4,Lemma 3.1]). Define A to be the linear span in F/I of all words x ∈ W \{0} satisfying bal(x) = 0.…”

“…′ is a commutative unital locally matricial algebra, and the system S satisfies property (C) of [4]. (The accurate assertion of (C) is here unimportant, as we will only need it to apply a lemma of [4]).…”

“…(The accurate assertion of (C) is here unimportant, as we will only need it to apply a lemma of [4]). If we can show that…”

“…Based on the work of Cuntz and Krieger in [7], in [4] the first named author considered a class of so-called Cuntz-Krieger type algebras relying on a flexible generators and relations approach. This class, which is recalled in Section 2, includes (aperiodic) Cuntz-Krieger algebras [7], higher rank Exel-Laca algebras [5], (aperiodic) higher rank graph C * -algebras [8,9], (aperiodic) ultragraph algebras [13] and (cancelling) higher rank semigraph C * -algebras [2].…”