Abstract
This paper is a continuation of the paper, Matsumoto [‘Subshifts,
$\lambda $
-graph bisystems and
$C^*$
-algebras’, J. Math. Anal. Appl. 485 (2020), 123843]. A
$\lambda $
-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift
$\Lambda $
, there exists a
$\lambda $
-graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra
${\mathcal {F}}_{\mathcal {L}}$
with shift automorphism
$\rho _{\mathcal {L}}$
from a
$\lambda $
-graph bisystem
$({\mathcal {L}}^-,{\mathcal {L}}^+)$
, and define a
$C^*$
-algebra
${\mathcal R}_{\mathcal {L}}$
by the crossed product
. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If
$\lambda $
-graph bisystems come from two-sided subshifts, these
$C^*$
-algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the
$C^*$
-algebra
${\mathcal R}_{\mathcal {L}}$
and the K-theory formulas of the
$C^*$
-algebras
${\mathcal {F}}_{\mathcal {L}}$
and
${\mathcal R}_{\mathcal {L}}$
. The K-group for the AF-algebra
${\mathcal {F}}_{\mathcal {L}}$
is regarded as a two-sided extension of the dimension group of subshifts.
EVERY SHIFT AUTOMORPHISM VARIETY HAS AN INFINITE SUBDIRECTLY IRREDUCIBLE MEMBER
