Intuition on the Cuntz-Krieger relations for Leavitt path algebras and graph $C^*$-algebras Let $k$ be a commutative ring with unity and $E$ a quiver with source and target functions $E^1 \xrightarrow{s,t} E^0$. The Leavitt path algebra of $E$ is the quotient of the path algebra of the double graph $k(E \sqcup E^\ast)$ by the Cuntz-Krieger relations
$$
e^\ast f := \delta_{s,f} \ t(f) \tag{CK 1}
$$
for each pair of edges $e,f \in E^1$ and
$$
v = \sum_{e  \ : \ s(e) = v}\tag{CK 2}ee^\ast
$$
for each regular vertex $v \in E^0$. This means that $s^{-1}(v)$ is non-empty and finite, which means that $v$ is not a sink or an infinite emitter.
This is analogous to the construction of graph $C^*$-algebras, where the original Cuntz-Krieger relations arose, and include several algebras of interest. However, I would like to gain some intuition as to why these relations were considered in the first place.
The original paper of Cuntz seems to relate $(CK1)$ and $(CK2)$ to topological Markov chains and symbolic dynamics, but I've had a hard time following the motivation. I'd really appreciate an explanation for non-experts.
 A: If $A$ is an $n\times n$ matrix of zeros and ones,  the Markov space
$$
  \Sigma _A=\big\{x=(x_k)_{k=1}^\infty \in  \{1, 2, \ldots , n\}^\mathbb N: A_{x_k, x_{k+1}}=1, \text{ for all } k\in \mathbb N\big\}
  $$
may be partitioned by the so called cylinder sets
$$
  Z_i = \big\{x=(x_k)_k\in  \Sigma _A: x_1=i\big\},
  $$
defined for each $i$ in $\{1, 2, \ldots , n\}$.
The shift map
$$
  S: (x_1, x_2, x_3, \ldots ) \mapsto  (x_2, x_3, x_4, \ldots )
  $$
maps $\Sigma _A$ to itself, and it is evidently injective when restricted to each cylinder $Z_i$.  We may then consider the
inverse branches of the shift, namely the maps
$$
  \theta _i : (x_1, x_2, \ldots ) \in    S(Z_i) \mapsto    (i,  x_1, x_2, \ldots ) \in   \Sigma _A.
  $$
Thus, while $S$ deletes the first coordinate of $x$, we see that $\theta _i$ inserts $i$ as a first coordinate of $x$.
However insertion cannot always be done since  $(i, x_1, x_2, \ldots )$ may sometimes fall outside $\Sigma _A$.
In fact notice that the
domain of each $\theta _i$ turns out to be
$$
  S(Z_i) = \bigcup_{{\buildrel {1\leq j\leq n} \over {A_{i, j}=1}}} Z_j.
  \tag 1
  $$
On the other hand  the  range of $\theta _i$ is precisely $Z_i$.
The original Cuntz-Krieger paper may be considered as a quantization of the Markov shift, where  $\theta _1, \theta _2, \ldots , \theta _n$ are replaced
by partial isometric operators  $S_1, S_2, \ldots , S_n$  on a Hilbert space.   The fact that the range  projections
$$
  P_i:= S_iS_i^*
  $$
are
required to add up to one may be seen as an interpretation of the fact that  $\Sigma _A$ is partitioned by the ranges of the
$\theta _i$.
The Cuntz-Krieger relation
$$
  S_i^*S_i = \sum_{j=1}^n A_{i,j} P_j,
  $$
if written in the equivalent form
$$
  S_i^*S_i = \sum_{{\buildrel {1\leq j\leq n} \over {A_{i, j}=1}}}  P_j,
  $$
expresses that the source projections   $S_i^*S_i$ behaves somewhat like the domains of the $\theta _i$, as  in (1).
It is therefore perhaps unsurprising that one of the main early results about the Cuntz-Krieger algebras  is that two Markov shifts are
conjugate iff the corresponding Cuntz-Krieger algebras are isomorphic.
