Bipartite Graph C*-Algebras

• Bipartite graph C*-algebras are C*-algebras constructed from directed bipartite graphs with a canonical parity functor that distinguishes edge origins.
• The grading automorphism, determined by the bipartition, twists the standard Cuntz–Krieger relations, yielding graded K-theoretic invariants through a signed adjacency matrix.
• For complete bipartite graphs, constructing tile complexes and 2-rank graph structures produces higher-rank C*-algebras with explicitly computable K-theory and homology.

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 (Kumjian et al., 2017, Mutter, 2020).

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 $\mathbb Z_2$0 is formed by taking $\mathbb Z_2$1, $\mathbb Z_2$2, 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 $\mathbb Z_2$3 paths occur at the generating level (Kumjian et al., 2017).

The bipartition gives a canonical parity functor

$\mathbb Z_2$4

defined by

$\mathbb Z_2$5

and

$\mathbb Z_2$6

Equivalently, all edges leaving the “odd” side $\mathbb Z_2$7 are declared odd, and those leaving $\mathbb Z_2$8 are declared even. Since any path factors uniquely into length-one edges, this extends to a functor on all of $\mathbb Z_2$9.

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 $2$0-graph case one may take the usual graph C*-algebra $2$1 and equip it with the grading automorphism $2$2 determined by $2$3. Concretely, $2$4 is the universal C*-algebra generated by

$2$5

subject to the usual Cuntz–Krieger relations

$2$6

$2$7

$2$8

The grading automorphism is given on generators by

$2$9

$2$0

so vertices are even, while an edge is even or odd according to whether $2$1 or $2$2 (Kumjian et al., 2017).

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 $2$3 over $2$4 acquires a grading operator $2$5 coming from $2$6, 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, $2$7 is trivially graded, so

$2$8

One therefore obtains the short exact sequence

$2$9

and hence

$1$0

$1$1

Equivalently, if one writes a column vector of formal generators indexed by $1$2, the boundary map sending that vector to $1$3 gives precisely these kernels and cokernels (Kumjian et al., 2017).

If

$1$4

then the signed adjacency matrix decomposes as

$1$5

where

$1$6

Accordingly,

$1$7

acts on $1$8.

The comparison with ordinary graph K-theory is immediate. In the ungraded theory one uses

$1$9

so that

$V=V_0\sqcup V_1$0

The grading replaces $V=V_0\sqcup V_1$1 by $V=V_0\sqcup V_1$2 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=V_0\sqcup V_1$3 back to $V=V_0\sqcup V_1$4 and so “sees” the bipartition in an essential way. Thus the graded K-theory refines the ordinary K-theory by detecting an additional $V=V_0\sqcup V_1$5-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

$V=V_0\sqcup V_1$6

on $V=V_0\sqcup V_1$7 “white” vertices $V=V_0\sqcup V_1$8 and $V=V_0\sqcup V_1$9 “black” vertices $E\subset V\times V$0, with one undirected edge $E\subset V\times V$1 for each $E\subset V\times V$2, $E\subset V\times V$3. Using a result of Vdovina, one can build a connected $E\subset V\times V$4-dimensional CW complex $E\subset V\times V$5, called the tile complex, all of whose vertices have link isomorphic to $E\subset V\times V$6 (Mutter, 2020).

The construction introduces two sets

$E\subset V\times V$7

$E\subset V\times V$8

each equipped with the involution $E\subset V\times V$9. For each edge $V_0$0 of $V_0$1, one forms four pointed square faces

$V_0$2

each oriented counter-clockwise from a distinguished base-vertex. Gluing these squares along like-labelled and like-oriented edges yields $V_0$3. The set of pointed tiles is denoted

$V_0$4

and the set of corresponding unpointed squares is denoted $V_0$5.

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 $V_0$6-graph; instead it becomes the local link structure of a $V_0$7-dimensional complex, permitting higher-rank graph realizations of the same underlying combinatorics.

5. Two commuting $V_0$8-rank-graph structures and their universal C*-algebras

A $V_0$9-rank graph $V_1$0 is a small category $V_1$1 together with a functor

$V_1$2

satisfying the factorisation property: if $V_1$3 and $V_1$4, then there are unique $V_1$5 with $V_1$6, $V_1$7, and $V_1$8. For the tile complex of a complete bipartite graph, two distinct commuting $V_1$9-rank-graph structures are defined, one on pointed tiles and one on unpointed tiles (Mutter, 2020).

In the pointed case, one defines two $1$0 matrices $1$1 indexed by $1$2. If

$1$3

then

$1$4

$1$5

and zero otherwise. These matrices are symmetric, each row and column has at least one nonzero entry, and

$1$6

Moreover, the Unique Common Extension property holds: whenever $1$7 and $1$8, there is a unique $1$9 with $\mathbb Z_2$00. The standard Kumjian–Pask commuting-matrix construction then yields a $\mathbb Z_2$01-rank graph $\mathbb Z_2$02 with $\mathbb Z_2$03. The same construction on $\mathbb Z_2$04, using

$\mathbb Z_2$05

gives the unpointed $\mathbb Z_2$06-rank graph $\mathbb Z_2$07.

For any row-finite $\mathbb Z_2$08-rank graph with no sources and finite vertex set $\mathbb Z_2$09, the higher-rank graph algebra $\mathbb Z_2$10 is the universal C*-algebra generated by partial isometries $\mathbb Z_2$11 satisfying the Cuntz–Krieger relations: $\mathbb Z_2$12

$\mathbb Z_2$13

$\mathbb Z_2$14

$\mathbb Z_2$15

In particular, one may form $\mathbb Z_2$16 and $\mathbb Z_2$17 from the two rank-graph realizations of the same tile complex.

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

For finite, row-finite $\mathbb Z_2$18-graphs, a spectral-sequence calculation of Evans gives the K-groups in terms of the adjacency matrices $\mathbb Z_2$19: $\mathbb Z_2$20 where

$\mathbb Z_2$21

In the pointed tile-complex case one takes $\mathbb Z_2$22, $\mathbb Z_2$23. A direct generators-and-relations argument shows that

$\mathbb Z_2$24

where $\mathbb Z_2$25 and $\mathbb Z_2$26. In every case,

$\mathbb Z_2$27

with

$\mathbb Z_2$28

The same pattern holds for the unpointed $\mathbb Z_2$29-rank graph $\mathbb Z_2$30. For $\mathbb Z_2$31,

$\mathbb Z_2$32

together with special degenerations when $\mathbb Z_2$33 or $\mathbb Z_2$34.

The tile complex also has computable ordinary homology. After collapsing one chosen square to a point, one obtains a $\mathbb Z_2$35-dimensional CW complex with one $\mathbb Z_2$36-cell, $\mathbb Z_2$37 loops in dimension $\mathbb Z_2$38, and $\mathbb Z_2$39 $\mathbb Z_2$40-cells. One finds

$\mathbb Z_2$41

$\mathbb Z_2$42

and

$\mathbb Z_2$43

Although the Evans spectral sequence is not literally the cellular chain complex of $\mathbb Z_2$44, there is a well-known intimate relation between the ranks of $\mathbb Z_2$45 and the ranks $\mathbb Z_2$46 appearing in the K-groups.

The same combinatorial scheme extends to even $\mathbb Z_2$47-gon complexes built from Vdovina’s construction, and to more general systems of $\mathbb Z_2$48-gons glued so that each link is $\mathbb Z_2$49. One defines horizontal and vertical adjacency of pointed $\mathbb Z_2$50-gons by reflecting across the appropriate axis through midpoints of opposite sides; the resulting $\mathbb Z_2$51- and $\mathbb Z_2$52-matrices again commute and satisfy UCE, yielding $\mathbb Z_2$53-rank graphs $\mathbb Z_2$54 and $\mathbb Z_2$55. In the pointed, even-$\mathbb Z_2$56 case,

$\mathbb Z_2$57

whereas in the unpointed case the K-groups are independent of $\mathbb Z_2$58 and coincide with those of $\mathbb Z_2$59.

A further invariant is the class of the identity in $\mathbb Z_2$60, which has finite order equal to $\mathbb Z_2$61, or half that in the unpointed variant if $\mathbb Z_2$62 is even. Hence, for $\mathbb Z_2$63, 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 $\mathbb Z_2$64. 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.