C ∗ -algebras arising from Dyck systems of topological Markov chains
Abstract
Let
A
be an
N×N
irreducible matrix with entries in
{0,1}
. We define the topological Markov Dyck shift
D
A
to be a nonsofic subshift consisting of the
2N
brackets
(
1
,...,
(
N
,
)
1
,...,
)
N
with both standard bracket rule and Markov chain rule coming from
A
. The subshift is regarded as a subshift defined by the canonical generators
S
∗
1
,...,
S
∗
N
,
S
1
,...,
S
N
of the Cuntz-Krieger algebra ${\Cal O}_A$. We construct an irreducible
λ
-graph system
L
Ch(
D
A
)
that presents the subshift
D
A
so that we have an associated simple purely infinite
C
∗
-algebra ${\Cal O}_{{\frak L}^{Ch(D_A)}}$. We prove that ${\Cal O}_{{\frak L}^{Ch(D_A)}}$ is a universal unique
C
∗
-algebra subject to some operator relations among
2N
generating partial isometries. Some examples are presented such that they are not stably isomorphic to any Cuntz-Krieger algebra.