---
title: Bipartite Graph C*-Algebras
url: https://www.emergentmind.com/topics/bipartite-graph-c-algebras
type: topic
---

# Bipartite Graph C*-Algebras

Bipartite graph C*-algebras, in the sense relevant here, arise in two complementary frameworks. For directed bipartite graphs viewed as \(1\)-graphs, the bipartition determines a \(\mathbb Z_2\)-valued functor and hence a grading automorphism on the associated graph C*-algebra, so that graded K-theory is controlled by a signed adjacency matrix. For complete bipartite graphs, one may instead build a connected \(2\)-dimensional CW complex whose link at each vertex is the given graph, endow this complex with commuting \(2\)-rank-graph structures, and form higher-rank graph C*-algebras with explicit K-groups. In both settings, the bipartite combinatorics are encoded directly in the algebraic and K-theoretic invariants [1706.00563], [2004.11602].

## 1. Directed bipartite graphs as \(1\)-graphs

Let \(V=V_0\sqcup V_1\) be a finite partition of the vertices and let \(E\subset V\times V\) be the directed edges, each edge necessarily running from \(V_0\) to \(V_1\) or vice versa. The usual graph \(1\)-category \(\Lambda\) is formed by taking \(\Lambda^0=V\), \(\Lambda^1=E\), and composition by concatenation of paths. This places an ordinary directed bipartite graph inside the Kumjian–Pask formalism for higher-rank graphs, but in the present case only degree \(1\) paths occur at the generating level [1706.00563].

The bipartition gives a canonical parity functor
\[
c:\Lambda\to \mathbb Z_2
\]
defined by
\[
c(v)=0,\qquad v\in V,
\]
and
\[
c(e)=
\begin{cases}
0 & \text{if } s(e)\in V_0,\\
1 & \text{if } s(e)\in V_1,
\end{cases}
\qquad e\in E.
\]
Equivalently, all edges leaving the “odd” side \(V_1\) are declared odd, and those leaving \(V_0\) are declared even. Since any path factors uniquely into length-one edges, this extends to a functor on all of \(\Lambda\).

This construction isolates a structural feature specific to bipartite graphs: the parity is attached not to the geometric edge alone, but to the side of the bipartition from which the edge departs. A plausible implication is that the categorical path structure already remembers the asymmetry between the two parts of the bipartition before any C*-algebra is formed.

## 2. Grading the graph C*-algebra

In the \(1\)-graph case one may take the usual graph C*-algebra \(C^*(\Lambda)\) and equip it with the grading automorphism \(\alpha\) determined by \(c\). Concretely, \(C^*(\Lambda,c)\) is the universal C*-algebra generated by
\[
\{p_v: v\in V\},\qquad \{s_e: e\in E\},
\]
subject to the usual Cuntz–Krieger relations
\[
\text{(CK1)}\quad p_v \text{ orthogonal projections},\quad \sum_{r(e)=v} s_e s_e^* = p_v \ \text{(no sources)},
\]
\[
\text{(CK2)}\quad s_e^* s_e = p_{s(e)},
\]
\[
\text{(CK3)}\quad s_e p_{s(e)}=s_e=p_{r(e)} s_e.
\]
The grading automorphism is given on generators by
\[
\alpha(p_v)=p_v,
\]
\[
\alpha(s_e)=(-1)^{c(e)}s_e,
\]
so vertices are even, while an edge is even or odd according to whether \(c(e)=0\) or \(1\) [1706.00563].

This realizes the bipartite graph C*-algebra as a graded C*-algebra in the sense used in Kasparov theory. The grading is not an external decoration: it changes the correspondence picture and therefore changes the exact sequence used to compute K-theory. In particular, the graph bimodule \(X:=X(\Lambda)\) over \(A:=C_0(\Lambda^0)\) acquires a grading operator \(\omega\) coming from \(c\), placing the graph algebra inside the graded Pimsner framework.

## 3. Graded K-theory and the signed adjacency matrix

Corollary 4.5 in the graded Pimsner theory yields a six-term exact sequence in graded K-theory for the graph correspondence. In the bipartite graph situation, \(A=C_0(V)\cong \mathbb C^{|V|}\) is trivially graded, so
\[
K_1^{gr}(A)=0,\qquad K_0^{gr}(A)=\mathbb Z^{|V|}.
\]
One therefore obtains the short exact sequence
\[
0 \to K_1^{gr}(C^*(\Lambda,c)) \to \mathbb Z^{|V|}
\xrightarrow{\,1-(A_c)^t\,}
\mathbb Z^{|V|}
\to K_0^{gr}(C^*(\Lambda,c)) \to 0
\]
and hence
\[
K_1^{gr}(C^*(\Lambda,c)) \cong \ker\bigl(1-(A_c)^t\bigr),
\]
\[
K_0^{gr}(C^*(\Lambda,c)) \cong \operatorname{coker}\bigl(1-(A_c)^t\bigr).
\]
Equivalently, if one writes a column vector of formal generators indexed by \(V\), the boundary map sending that vector to \((I-(A_c)^t)\) gives precisely these kernels and cokernels [1706.00563].

If
\[
V_0=\{v_1,\dots,v_{n_0}\},\qquad V_1=\{w_1,\dots,w_{n_1}\},
\]
then the signed adjacency matrix decomposes as
\[
A_c=
\begin{pmatrix}
0 & B\\
-C & 0
\end{pmatrix},
\]
where
\[
B_{i,j}=|\{e:s(e)=v_i,\ r(e)=w_j\}|,
\qquad
C_{j,i}=|\{e:s(e)=w_j,\ r(e)=v_i\}|.
\]
Accordingly,
\[
1-(A_c)^t=
\begin{pmatrix}
I_{n_0} & C^t\\
-B^t & I_{n_1}
\end{pmatrix}
\]
acts on \(\mathbb Z^{n_0}\oplus \mathbb Z^{n_1}\).

The comparison with ordinary graph K-theory is immediate. In the ungraded theory one uses
\[
A=
\begin{pmatrix}
0 & B\\
C & 0
\end{pmatrix},
\]
so that
\[
K_1(C^*(\Lambda)) \cong \ker(1-A^t),\qquad
K_0(C^*(\Lambda)) \cong \operatorname{coker}(1-A^t).
\]
The grading replaces \(C\) by \(-C\) in the lower-left block. For many bipartite graphs, the sign-twist can change the rank or torsion in the resulting K-groups. From the point of view of Pimsner’s sequence, the grading picks up the parity of edges from \(V_1\) back to \(V_0\) and so “sees” the bipartition in an essential way. Thus the graded K-theory refines the ordinary K-theory by detecting an additional \(\mathbb Z_2\)-valued obstruction attached to the return-edges from the odd side.

## 4. Complete bipartite graphs and tile complexes

A second class of bipartite graph C*-algebras begins with the complete connected bipartite graph
\[
\kappa=\kappa(\alpha,\beta)
\]
on \(\alpha\) “white” vertices \(u_1,\dots,u_\alpha\) and \(\beta\) “black” vertices \(v_1,\dots,v_\beta\), with one undirected edge \(u_i v_j\) for each \(1\le i\le \alpha\), \(1\le j\le \beta\). Using a result of Vdovina, one can build a connected \(2\)-dimensional CW complex \(TC(\kappa)\), called the tile complex, all of whose vertices have link isomorphic to \(\kappa\) [2004.11602].

The construction introduces two sets
\[
U=\{u_i^1,u_i^2,\bar u_i^1,\bar u_i^2:1\le i\le \alpha\},
\]
\[
V=\{v_j^1,v_j^2,\bar v_j^1,\bar v_j^2:1\le j\le \beta\},
\]
each equipped with the involution \(x\mapsto \bar x\). For each edge \(u_i v_j\) of \(\kappa\), one forms four pointed square faces
\[
\begin{aligned}
A_{ij}&=[u_i^1,v_j^1,u_i^2,v_j^2],\\
B_{ij}&=[\bar u_i^1,\bar v_j^2,\bar u_i^2,\bar v_j^1],\\
C_{ij}&=[u_i^2,v_j^2,u_i^1,v_j^1],\\
D_{ij}&=[\bar u_i^2,\bar v_j^1,\bar u_i^1,\bar v_j^2],
\end{aligned}
\]
each oriented counter-clockwise from a distinguished base-vertex. Gluing these squares along like-labelled and like-oriented edges yields \(TC(\kappa)\). The set of pointed tiles is denoted
\[
\mathcal S=\mathcal S(\kappa)=\{A_{ij},B_{ij},C_{ij},D_{ij}\},
\]
and the set of corresponding unpointed squares is denoted \(\mathcal S'=\mathcal S'(\kappa)\).

This passage from a graph to a square complex is central. The complete bipartite graph is no longer used merely as adjacency data for a \(1\)-graph; instead it becomes the local link structure of a \(2\)-dimensional complex, permitting higher-rank graph realizations of the same underlying combinatorics.

## 5. Two commuting \(2\)-rank-graph structures and their universal C*-algebras

A \(2\)-rank graph \((\Lambda,d)\) is a small category \(\Lambda\) together with a functor
\[
d:\Lambda\to \mathbb N^2
\]
satisfying the factorisation property: if \(d(\lambda)=(m_1,m_2)\) and \((m_1,m_2)=(n_1,n_2)+(p_1,p_2)\), then there are unique \(\mu,\nu\) with \(d(\mu)=(n_1,n_2)\), \(d(\nu)=(p_1,p_2)\), and \(\lambda=\nu\mu\). For the tile complex of a complete bipartite graph, two distinct commuting \(2\)-rank-graph structures are defined, one on pointed tiles and one on unpointed tiles [2004.11602].

In the pointed case, one defines two \(4\alpha\beta\times 4\alpha\beta\) matrices \(M_1,M_2\) indexed by \(\mathcal S\). If
\[
A=[x_1,y_1,x_2,y_2],\qquad B=[x_3,y_3,x_4,y_4],
\]
then
\[
M_1(A,B)=1 \iff y_1=\bar y_4 \text{ and } x_1\neq \bar x_3,
\]
\[
M_2(A,B)=1 \iff x_2=\bar x_3 \text{ and } y_1\neq \bar y_3,
\]
and zero otherwise. These matrices are symmetric, each row and column has at least one nonzero entry, and
\[
M_1M_2=M_2M_1.
\]
Moreover, the Unique Common Extension property holds: whenever \(M_1(A,B)=1\) and \(M_2(A,C)=1\), there is a unique \(D\in\mathcal S\) with \(M_2(B,D)=M_1(C,D)=1\). The standard Kumjian–Pask commuting-matrix construction then yields a \(2\)-rank graph \(\Lambda(\kappa)\) with \(\Lambda^0=\mathcal S\). The same construction on \(\mathcal S'\), using
\[
M_i'(A',B')=1 \iff \exists \text{ lifts } A\in A',\ B\in B' \text{ with } M_i(A,B)=1,
\]
gives the unpointed \(2\)-rank graph \(\Lambda'(\kappa)\).

For any row-finite \(2\)-rank graph with no sources and finite vertex set \(\Lambda^0\), the higher-rank graph algebra \(C^*(\Lambda)\) is the universal C*-algebra generated by partial isometries \(\{s_\lambda:\lambda\in\Lambda\}\) satisfying the Cuntz–Krieger relations:
\[
\text{(CK1)}\quad s_v \text{ are mutually orthogonal projections for } v\in\Lambda^0,
\]
\[
\text{(CK2)}\quad s_\mu s_\nu=s_{\mu\nu}\ \text{whenever } r(\nu)=s(\mu),
\]
\[
\text{(CK3)}\quad s_\lambda^*s_\lambda=s_{s(\lambda)},
\]
\[
\text{(CK4)}\quad
s_v=\sum_{\{\lambda\in \Lambda^{\mathbf n}:r(\lambda)=v\}} s_\lambda s_\lambda^*
\quad\text{for every } v\in\Lambda^0,\ \mathbf n\in\mathbb N^2.
\]
In particular, one may form \(C^*(\Lambda(\kappa))\) and \(C^*(\Lambda'(\kappa))\) from the two rank-graph realizations of the same tile complex.

## 6. K-theory, homology, higher polygons, and classification

For finite, row-finite \(2\)-graphs, a spectral-sequence calculation of Evans gives the K-groups in terms of the adjacency matrices \(M_E,M_F\):
\[
\begin{aligned}
K_0(C^*(\Lambda))
&\cong \mathbb Z^{r_0}\oplus \operatorname{tors}\!\Bigl(\operatorname{coker}(I-M_E^T,\ I-M_F^T)\Bigr),\\
K_1(C^*(\Lambda))
&\cong \mathbb Z^{r_1}\oplus \operatorname{tors}\!\Bigl(\operatorname{coker}(I-M_E,\ I-M_F)\Bigr),
\end{aligned}
\]
where
\[
r_0=r_1=
\rk\!\bigl(\operatorname{coker}(I-M_E^T,I-M_F^T)\bigr)
+\rk\!\bigl(\operatorname{coker}(I-M_E,I-M_F)\bigr).
\]
In the pointed tile-complex case one takes \(M_E=M_1\), \(M_F=M_2\). A direct generators-and-relations argument shows that
\[
\operatorname{coker}(I-M_1^T,I-M_2^T)\cong
\begin{cases}
\mathbb Z^8,&(\alpha,\beta)=(2,2),\\[4pt]
(\mathbb Z/b)^2\oplus \mathbb Z^{4(b+1)},&(\alpha,\beta)=(2,b+2),\\[4pt]
(\mathbb Z/a)^{\,b-a}\oplus(\mathbb Z/ab)^{\,a+1}\oplus\mathbb Z^{2(a+1)(b+1)},&a,b\ge 2,\ \gcd(a,b)=1,\\[4pt]
(\mathbb Z/a)^{\,b-a}\oplus(\mathbb Z/\operatorname{lcm}(a,b))^{\,a+1}\oplus(\mathbb Z/\gcd(a,b))^{\,a+2}\oplus\mathbb Z^{2(a+1)(b+1)},&a,b\ge 2,
\end{cases}
\]
where \(a=\alpha-2\) and \(b=\beta-2\). In every case,
\[
K_0(C^*(\Lambda(\kappa)))\cong K_1(C^*(\Lambda(\kappa)))
\cong \operatorname{coker}(I-M_1^T,I-M_2^T)\oplus \mathbb Z^r,
\]
with
\[
r=\rk(\operatorname{coker}(I-M_1^T,I-M_2^T)).
\]

The same pattern holds for the unpointed \(2\)-rank graph \(\Lambda'(\kappa)\). For \(a,b\ge 1\),
\[
K_\ast(C^*(\Lambda'(\kappa)))
\cong (\mathbb Z/2)^{(a+1)(b+1)-1}\oplus (\mathbb Z/2\gcd(a,b)),
\qquad \ast=0,1,
\]
together with special degenerations when \(\alpha=2\) or \(\beta=2\).

The tile complex also has computable ordinary homology. After collapsing one chosen square to a point, one obtains a \(2\)-dimensional CW complex with one \(0\)-cell, \(4\alpha+4\beta\) loops in dimension \(1\), and \(\alpha\beta+(\alpha-1)+(\beta-1)\) \(2\)-cells. One finds
\[
H_1(TC(\kappa))\cong \mathbb Z^{\alpha+\beta-2},
\]
\[
H_2(TC(\kappa))\cong \mathbb Z^{(\alpha-1)(\beta-1)},
\]
and
\[
H_n=0\qquad (n\ne 1,2).
\]
Although the Evans spectral sequence is not literally the cellular chain complex of \(TC(\kappa)\), there is a well-known intimate relation between the ranks of \(H_i\) and the ranks \(r_0,r_1\) appearing in the K-groups.

The same combinatorial scheme extends to even \(2t\)-gon complexes built from Vdovina’s construction, and to more general systems of \(2t\)-gons glued so that each link is \(\kappa\). One defines horizontal and vertical adjacency of pointed \(2t\)-gons by reflecting across the appropriate axis through midpoints of opposite sides; the resulting \(U\)- and \(V\)-matrices again commute and satisfy UCE, yielding \(2\)-rank graphs \(\Lambda_t(\kappa)\) and \(\Lambda_t'(\kappa)\). In the pointed, even-\(t\) case,
\[
K_\ast(C^*(\Lambda_t(\kappa)))
\cong \bigl(K_\ast(C^*(\Lambda(\kappa)))\bigr)^{\oplus t/2},
\qquad \ast=0,1,
\]
whereas in the unpointed case the K-groups are independent of \(t\) and coincide with those of \(\Lambda'(\kappa)\).

A further invariant is the class of the identity in \(K_0\), which has finite order equal to \(\gcd(\alpha-2,\beta-2)\), or half that in the unpointed variant if \(g\) is even. Hence, for \(\alpha,\beta\ge 3\), the Kirchberg–Phillips theorem implies that all these higher-rank-graph C*-algebras are simple, purely infinite, nuclear, and classified up to isomorphism by the K-theory data together with the position of \([1]\in K_0\). In this regime, the complete bipartite graph governs not only the combinatorics of the underlying square complex, but also the stable classification data of the associated C*-algebras.

Source: https://www.emergentmind.com/topics/bipartite-graph-c-algebras