Graph Traces on Product Graphs
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1 Graph Traces on Product Graphs Amy Isvik, Nate Kolo, Maria Ross Kansas State University - SUMaR REU July 6, 06 This work was carried out at the Kansas State University SUMaR program under support of NSF Grant # DSM-6877 and advised by Dr. Danny Crytser. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
2 Overview Directed graphs, product graphs, and their traces Higher-rank graphs and their traces 3 Products of higher rank graphs and their traces Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
3 Directed Graphs and Traces Definition Let E be a directed graph. A function g : E 0! [0, ] is a graph trace if (i) For any regular vertex v E 0, g(v) = X ee,r(e)=v g(s(e)). (ii) For any infinite receiver v E 0 and any finite collection of edges in r (v), we have nx g(v) g(s(e i )). (iii) X i= ve 0 g(v) = Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 3 / 33
4 Examples of Directed Graph Traces g(u) =t + t e e f g(v) = g(w) = g(v) =t g(w) =t g(v) = k k+ Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
5 Definition An extreme graph trace is a graph trace which cannot be written as a convex combination of other graph traces. That is, if g is an extreme graph trace and g = tg 0 +( t)g 00 for graph traces g 0, g 00 and t (0, ), then g 0 = g 00 = g. e f e f e f 0 0 Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 5 / 33
6 Theorem [Johnson] Let E be a finite directed graph with no cycles. Then there exists a bijection between the set of all sources of E, S E, and the set of all extreme traces on E, ext(t (E)). v e f v v Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 6 / 33
7 Box Product v e v v w w e v w w v e e w v e e w v w v w Box Product of Directed Graphs The box (Cartesian) product of E with F is the graph E F =(E 0 F 0, (E F 0 ) [ (E 0 F ), r, s ), where r, s are defined as follows: For all e E, f F, u E 0, v F 0 : r (e, v) =(r E (e), v) r (u, f )=(u, r F (f )) s (e, v) =(s E (e), v) s (u, f )=(u, s F (f )) Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 7 / 33
8 Box Product This product operation does not guarantee that the product of graph traces on factor graphs is a trace on the product graph. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 8 / 33
9 Tensor Product v e v v w w e v w w e e v w v w Tensor Product of Directed Graphs The tensor product of E with F is the graph E F =(E 0 F 0, E F, r, s ), such that for all (e, f ) E F we define: r (e, f )=(r E (e), r F (f )) and s (e, f )=(s E (e), s F (f )). Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 9 / 33
10 Tensor Product The tensor product of traces on factor graphs gives a trace on the product graph. This is not necessarily the only way to find traces on the product graph. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 0 / 33
11 Higher-rank graphs Definition A higher-rank graph, ork-graph,(, d), consists of a category and a degree functor d :! N k (i.e. d( )=d( )+d( )) satisfying the factorization property: forany, if d( )=m + n for m, n N k,then there exist unique µ, suchthat = µ and d(µ) = m,d( ) = n. For n N k,let n denote d (n) ={ :d( )=n}. v v v v 3 Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
12 Definition A k-graph is locally convex if whenever a vertex receives di erent colored edges, the sources of these edges also receive edges of each color other than the one it sends. v v w v v 3 w w Locally Convex Not Locally Convex Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
13 Definition Let be a (locally convex, row-finite) k-graph, and let 0 be its set of vertices. A function g : 0! [0, ] is called a higher-rank graph trace if (i) for any vertex v 0 and any degree n N k,wehave X v applen g(s( )) = g(v); (ii). X v 0 g(v) = Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 3 / 33
14 g(v )= 6 g(v )= 6 g(v )= 3 g(v 3 )= 3 Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
15 Definition Let E be a finite graph with no cycles and let v, w E 0.Then,definethe number of finite paths from v as n(v) = { E : s( = v}. Alsodefine the number of paths between v and w as n(v, w) = { E : s( )=v, r( )=w}. Theorem If is a finite locally convex k-graph with no cycles, then there is a one to one correspondence between sources and extreme traces defined by S 3 v 7! g v T ( ) where g v (w) = n(v,w) n(v). Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 5 / 33
16 w v Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 6 / 33
17 w g v (v) = Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 7 / 33
18 g v (w) = g v (v) = Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 8 / 33
19 g v (w) = 0 0 g v (v) = Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 9 / 33
20 Product of Higher Rank Graphs Definition Let be a k-graph and let be an `-graph. The product of and, denoted, is simply the Cartesian product of and, equipped with the following structure: (i) d(, ) =(d( ), d( )) (ii) r(, ) =(r( ), r( )) and likewise for the source map. (iii) (, )( 0, 0 )=( 0, 0 ) whenever both compositions in the factor graphs are defined. Note that with this degree map, the vertex set of isjust 0 0. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 0 / 33
21 vw w w f w f v w e w vf vf e w v w X vw vw v f v f v f v f v v e e v e w e w e w e w v w v w v w v w Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
22 t + t + t 3 + t t + t 3 t + t t + t t 3 + t t t t 3 t If you assign values to t, t, t 3, and t, then the graph trace values at the remaining vertices are as shown. Note that these graph traces must also satisfy the relation: t + t + t 3 + t =. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
23 Proposition Let g be a graph trace on and g be a graph trace on. Then g g (vw) =g (v)g (w) is a graph trace on. 8 8 X Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 3 / 33
24 0 8 8 X Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 / 33
25 Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 5 / 33
26 Proposition Let g be a graph trace on. Then define g : 0! [0, ] and g : 0! [0, ] by g (v) = X w 0 g(vw) g (w) = X v 0 g(vw). Then g is a graph trace on and g is a graph trace on. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 6 / 33
27 0 0 X Note: g (v )= P w 0 g(v w)= + = Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 7 / 33
28 Definition A graph trace g on a product higher-rank graph isaproduct trace if g = g g where g is a graph trace on and g is a graph trace on. g(v )= g(v )= g(v w )= g(v w )= g(w )= g(w )= g(v w )= g(v w )= Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 8 / 33
29 Extreme traces on the product graph can be understood using extreme traces on the factor graphs, as shown by these propositions. Proposition If g is a trace on andg is extreme, then g = g g. Proposition The product trace g g is extreme if and only if g and g are extreme. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 9 / 33
30 Conjecture Let and be higher-rank graphs. Then every extreme trace on is aproducttrace. Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, / 33
31 The End Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 3 / 33
32 Bibliography Jacob v.b. Hjelmborg. Purely infinite and stable C -algebras of graphs and dynamical systems. Ergodic Theory Dynam. Systems. : Matthew Johnson. The graph traces of finite graphs and applications to tracial states of C -algebras. New York Journal of Mathematics. : Ann Johnston and Andrew Reynolds. C -algebras of Graph Products. REU Report. Canisius College, 009. Richard Kadison and John Ringrose. Fundamentals of the Theory of Operator Algebras. American Mathematical Society. Graduate Studies in Mathematics. Vol David Pask and Adam Rennie. The noncommutative geometry of higher-rank graph C -algebras I: The index theorem. J.Funct.Anal. 33: Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, 06 3 / 33
33 Bibliography Isaac Namioka and R.R. Phelps. Tensor products of compact convex sets. Pac. J. Math. 3: Iain Raeburn. Graph Algebras. American Mathematical Society. CMBS Lecture Notes Iain Raeburn, Aidan Sims, and Trent Yeend. Higher-rank graphs and their C -algebras. Proc. Edin. Math. Soc. 6: Mark Tomforde. The ordered K 0 -group of a graph C -algebra. C.R. Math. Acad. Sci. Soc. 5: Amy Isvik, Nate Kolo, Maria Ross (SUMaR) Graph Traces July 6, / 33
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