Introduction
A directe d grap h i s a combinatoria l objec t consistin g o f vertice s an d oriente d
edges joinin g pair s o f vertices . W e ca n represen t suc h a grap h b y operator s o n
a Hilber t spac e H: th e vertice s ar e represente d b y mutuall y orthogona l close d
subspaces, o r mor e precisel y th e projection s ont o thes e subspaces , an d th e edge s
by operator s betwee n th e appropriat e subspaces . Th e grap h algebr a is , loosel y
speaking, th e C*-subalgebr a o f B(H) generate d b y thes e operators .
When the graph is finite and highl y connected, the graph algebra s coincide with
a family o f C*-algebras first studie d b y Cuntz an d Krieger i n 1980 [16]. Th e Cuntz -
Krieger algebras were quickly recognised to be a rich supply of examples for operato r
algebraists, an d als o croppe d u p i n som e unexpecte d situation s [87 , 131]. I n th e
past te n year s ther e ha s bee n a grea t dea l o f interes t i n grap h algebra s associate d
to infinit e graphs , an d thes e hav e arise n i n ne w contexts : i n non-abelia n dualit y
[83, 24], as deformations of commutative algebras [56, 57, 58], in non-commutativ e
geometry [12], and a s model s fo r th e classificatio n o f simpl e C*-algebra s [136].
Graph algebra s hav e a n attractiv e structur e theor y i n whic h algebrai c proper -
ties o f th e algebr a ar e relate d t o combinatoria l propertie s o f path s i n th e directe d
graph. Th e fundamenta l theorem s o f th e subjec t ar e analogue s o f those prove d b y
Cuntz an d Krieger, an d include a uniqueness theorem an d a description of the ideals
in graph algebras . Bu t w e now know much more: jus t abou t an y C*-algebraic prop -
erty a grap h algebr a migh t hav e ca n b e determine d b y lookin g a t th e underlyin g
graph.
Our goal s her e ar e t o describ e th e structur e theor y o f grap h algebras , an d t o
discuss tw o particularly promisin g extension s o f that theor y involvin g th e topolog -
ical graph s o f Katsura [73 ] an d th e higher-ran k graph s o f Kumjia n an d Pas k [81].
We provide ful l proof s o f the fundamenta l theorems , an d als o when w e think som e
insight ca n b e gaine d b y provin g a specia l cas e o f a publishe d resul t o r b y takin g
an alternativ e rout e t o it . Otherwis e w e concentrat e o n describin g th e mai n idea s
and givin g reference s t o th e literature .
Outline. Th e cor e materia l i s i n th e first fou r chapters , wher e w e discuss th e
uniqueness theorem s an d th e idea l structure . Thes e theorem s wer e first prove d fo r
infinite graph s b y realising the graph algebr a a s the C*-algebra o f a locally compac t
groupoid, an d applyin g result s o f Renaul t [83 , 82] . Ther e ar e no w severa l othe r
approaches t o thi s material ; th e elementar y method s w e us e her e ar e base d o n th e
original argument s o f Cunt z an d Krieger , bu t incorporat e severa l simplification s
which hav e bee n mad e ove r th e years . Thes e technique s w
Tork
bes t fo r th e row -
finite graph s i n whic h eac h verte x receive s jus t finitely man y edges : i n Chapte r 5
we describ e a metho d o f Drine n an d Tomford e fo r reducin g problem s t o th e row -
finite case .
l
http://dx.doi.org/10.1090/cbms/103/01
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