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MR2413313 (Review) 46L05 (37B99 46L80)

Katsura, Takeshi (J-KEIO); Muhly, Paul S. (1-IA) ; Sims, Aidan (5-WLG-SMS);Tomforde, Mark (1-HST)Ultragraph C∗-algebras via topological quivers. (English summary)Studia Math.187(2008),no. 2,137–155.

In this paper the authors show that the ultragraph algebrasC∗(G) introduced in [M. Tomforde, J.Operator Theory50(2003), no. 2, 345–368;MR2050134 (2005b:46128)] may be realised as topo-logical graphs in the sense of [T. Katsura, Trans. Amer. Math. Soc.356(2004), no. 11, 4287–4322(electronic);MR2067120 (2005b:46119)] or topological quivers in the sense of [P. S. Muhly andM. Tomforde, Internat. J. Math.16(2005), no. 7, 693–755;MR2158956 (2006i:46099)]. An ultra-graph may be viewed as a generalisation of a directed graph in which the source of an edge may beset-valued. The connections established enable them to plug into the results of [T. Katsura, Trans.Amer. Math. Soc.356(2004), no. 11, 4287–4322 (electronic);MR2067120 (2005b:46119); P. S.Muhly and M. Tomforde, Internat. J. Math.16(2005), no. 7, 693–755;MR2158956 (2006i:46099)]to study theK-theory and gauge-invariant ideal structure of ultragraphC∗-algebras.

The formulas for theK-groups are given in Theorem 5.4, whereK0(C∗(G)) is proved to beisomorphic to the cokernel of a certain map andK1(C∗(G)) is proved to be isomorphic to thekernel of that map.

In Section 6 collections of subsets of the vertex set which parametrise the gauge-invariant idealsof an ultragraphC∗-algebra are studied. By analogy with the standard theory of graphC∗-algebras[see, e.g., A. Kumjian et al., J. Funct. Anal.144(1997), no. 2, 505–541;MR1432596 (98g:46083)]these collections are called saturated, hereditary subsets ofG0, the collection of subsets of the vertexset of the ultragraph which give rise to projections inC∗(G).

In Section 6 it is shown that there is a lattice isomorphism between the lattices of saturatedhereditary subsets ofG0 and of the gauge-invariant ideals ofC∗(G). In Section 7, the authors arriveat a condition (K) (analogous to condition (K) from [A. Kumjian et al., J. Funct. Anal.144(1997),no. 2, 505–541;MR1432596 (98g:46083)]), which characterises the ultragraphsG for which everyideal ofC∗(G) is gauge-invariant.

Reviewed byTeresa Gay Bates

References

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2. J. Cuntz and W. Krieger,A class ofC∗-algebras and topological Markov chains, Invent. Math.56 (1980), 251–268.MR0561974 (82f:46073a)

3. R. Exel and M. Laca,Cuntz-Krieger algebras for infinite matrices, J. Reine Angew. Math. 512(1999), 119–172.MR1703078 (2000i:46064)

4. C. Farthing, P. S. Muhly and T. Yeend,Higher-rank graphC∗-algebras: an inverse semigroup

and groupoid approach, Semigroup Forum 71 (2005), 159–187.MR2184052 (2006h:46052)5. N. J. Fowler, M. Laca and I. Raeburn,TheC∗-algebras of infinite graphs, Proc. Amer. Math.

Soc. 128 (2000), 2319–2327.MR1670363 (2000k:46079)6. T. Katsura,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-

algebras I, fundamental results, Trans. Amer. Math. Soc. 356 (2004), 4287–4322.MR2067120(2005b:46119)

7. T. Katsura,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras III, ideal structures, Ergodic Theory Dynam. Systems 26 (2006), 1805–1854.MR2279267 (2007j:46090)

8. A. Kumjian, D. Pask and I. Raeburn,Cuntz-Krieger algebras of directed graphs, Pacific J.Math. 184 (1998), 161–174.MR1626528 (99i:46049)

9. A. Kumjian, D. Pask, I. Raeburn and J. Renault,Graphs, groupoids, and Cuntz-Krieger alge-bras, J. Funct. Anal. 144 (1997), 505–541.MR1432596 (98g:46083)

10. A. Marrero and P. S. Muhly,Groupoid and inverse semigroup presentations of ultragraphC∗-algebras, Semigroup Forum, to appear.cf. MR2457327

11. P. S. Muhly and M. Tomforde,Topological Quivers, Internat. J. Math. 16 (2005), 693–755.MR2158956 (2006i:46099)

12. A. L. T. Paterson,Graph inverse semigroups, groupoids and theirC∗-algebras, J. OperatorTheory 48 (2002), 645–662.MR1962477 (2004h:46066)

13. M. Tomforde,A unified approach to Exel-Laca algebras andC∗-algebras associated to graphs,ibid. 50 (2003), 345–368.MR2050134 (2005b:46128)

14. M. Tomforde,Simplicity of ultragraph algebras, Indiana Univ. Math. J. 52 (2003), 901–926.MR2001938 (2004f:46082)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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MR2414327 (Review) 46L55 (37A20)

Katsura, Takeshi (J-TOKYO); Matui, Hiroki (J-CHIBEGS)Classification of uniformly outer actions ofZ2 on UHF algebras. (English summary)Adv. Math.218(2008),no. 3,940–968.

In the study of automorphisms and group actions on operator algebras, one of the most importantand useful properties is the Rohlin property.

In [J. Math. Soc. Japan51 (1999), no. 3, 583–612;MR1691489 (2000f:46085)], H. Nakamurashowed that the uniform outerness of actions ofZn on UHF algebras is equivalent to the Rohlinproperty for actions, and obtained the classification of product type actions ofZ2 on UHF algebras.

In the paper under review, the authors extend Nakamura’s results and classify uniformly outeractions ofZ2 on UHF algebras byK-theoretical invariants. In particular, for a UHF algebra ofinfinite type, invariants are automatically trivial, and hence any two uniformly outer actions ofZ2

are cocycle conjugate.One of the technically difficult parts of the authors’ work is the construction of a homotopy of

unitaries. They introduce the notion of admissible cocycles, and solve this problem. Together withthe Rohlin property, they show the cohomology vanishing theorem. Then by applying the Evans-Kishimoto intertwining argument [D. E. Evans and A. Kishimoto, Hokkaido Math. J.26 (1997),no. 1, 211–224;MR1432548 (98e:46081)], they classify actions.

Reviewed byToshihiko Masuda

References

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4. D.E. Evans, A. Kishimoto, Trace scaling automorphisms of certain stable AF algebras,Hokkaido Math. J. 26 (1997) 211–224.MR1432548 (98e:46081)

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8. M. Izumi, The Rohlin property for automorphisms ofC∗-algebras, in: Mathematical Physicsin Mathematics and Physics, Siena, 2000, in: Fields Inst. Commun., vol. 30, Amer. Math. Soc.,Providence, RI, 2001, pp. 191–206.MR1867556 (2003j:46098)

9. M. Izumi, Finite group actions onC∗-algebras with the Rohlin property I, Duke Math. J. 122(2004) 233–280.MR2053753 (2005a:46142)

10. M. Izumi, Finite group actions onC∗-algebras with the Rohlin property II, Adv. Math. 184(2004) 119–160.MR2047851 (2005b:46153)

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13. A. Kishimoto, Unbounded derivations in AT algebras, J. Funct. Anal. 160 (1998) 270–311.MR1658684 (2000b:46113)

14. H. Lin, Approximate homotopy of homomorphisms fromC(X) into a simpleC∗-algebra,preprint, arXiv:math/0612125.

15. H. Matui, Classification of outer actions ofZN onO2, preprint.

16. H. Nakamura, The Rohlin property forZ2-actions on UHF algebras, J. Math. Soc. Japan 51(1999) 583–612.MR1691489 (2000f:46085)

17. H. Nakamura, Aperiodic automorphisms of nuclear purely infinite simpleC∗-algebras, ErgodicTheory Dynam. Systems 20 (2000) 1749–1765.MR1804956 (2002a:46089)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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MR2407845 (2009b:20005)20C05Katsura, Takeshi (J-HOKK)Permutation presentations of modules over finite groups. (English summary)J. Algebra319(2008),no. 9,3653–3665.

For a finite groupG, let ZG denote the group algebra ofG over the ringZ of integers. AZG-moduleM is called a permutation module if there is aZ-basis ofM which is permuted byG. Apermutation presentation of aZG-moduleM is an exact sequence0→ F → F →M → 0 withF a permutation module. The main result in the paper under review states that everyZG-modulehas a permutation presentation if and only if every Sylow-subgroup ofG is cyclic.

Reviewed byChangchang Xi

References

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6. T. Katsura, A construction of actions on Kirchberg algebras which induce given actions ontheirK-groups, J. Reine Angew. Math., in press.MR2400990 (2009b:46140)

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(2005m:13012)8. D.-S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959) 700–712.MR0104721 (21

#3474)9. D.J.S. Robinson, A Course in the Theory of Groups, second edition, Grad. Texts in Math., vol.

80, Springer-Verlag, New York, 1996.MR1357169 (96f:20001)10. J.-P. Serre, Local fields, translated from the French by Marvin Jay Greenberg, Grad. Texts in

Math., vol. 67, Springer-Verlag, New York/Berlin, 1979.MR0554237 (82e:12016)11. J. Spielberg, Non-cyclotomic presentations of modules and prime-order automorphisms of

Kirchberg algebras, J. Reine Angew. Math., in press.Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR2400990 (2009b:46140)46L80 (19K35 46L05 46L35 46L40 46L55)

Katsura, Takeshi (J-HOKK)A construction of actions on Kirchberg algebras which induce given actions on theirK-groups. (English summary)J. Reine Angew. Math.617(2008), 27–65.

A Kirchberg algebra is aC∗-algebra that is purely infinite, simple, separable, nuclear, and inthe UCT class. A remarkable theorem due to Kirchberg and Phillips shows that the Kirchbergalgebras are classified by theirK-theory, and furthermore, ifA is any Kirchberg algebra then anautomorphism on theK-theoryK∗(A) lifts to an automorphism onA.

In the paper under review the author considers actions of groups on theK-theory of a Kirchbergalgebra. In particular, the main result of the paper shows that ifA is a Kirchberg algebra, andΓ isa finite group all of whose Sylow subgroups are cyclic, then any action ofΓ onK∗(A) lifts to anaction ofΓ onA. Since any finite cyclic subgroupZ/nZ satisfies the hypothesis of this theorem,as a corollary one obtains that ifA is a Kirchberg algebra, then any automorphism ofK∗(A) liftsto an automorphism ofA with the same order.

The method of proof involves the construction of a KirchbergOA,B from two matricesA,B ∈MN(Z), and the construction of a groupΓA,B that acts onOA,B. TheC∗-algebraOA,B generalizesthe construction of the Cuntz-Krieger algebra, and in fact forA ∈Mn(N) it is the case thatOA,0

∼=OA andOA,A

∼= OA⊗C(T). The author gives an extensive analysis of theK-theory ofOA,B andthe action on theK-theory induced by the action ofΓA,B.

Throughout the paper the author does his best to make the proofs as self-contained as possible.The author points out that some of the lemmas and technical aspects of the analysis ofOA,B

contained in this paper follow from general results for topological graph algebras and Cuntz-

Pimsner algebras. Despite this, however, the author gives complete and direct proofs of theseresults for the sake of the reader. There are also two appendices; the first contains some technicalresults that can be used to give a concrete proof of a lemma obtained byKK-theory, and thesecond derives results not necessary for the proof of the main theorem but useful for consideringexamples.

Reviewed byMark Tomforde

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2. Blackadar, B.,K-theory for operator algebras, Second edition, Math. Sci. Res. Inst. Publ.5,Cambridge University Press, Cambridge 1998.MR1656031 (99g:46104)

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8. Exel, R., Laca, M.,TheK-theory of Cuntz-Krieger algebras for infinite matrices,K-Theory19(2000), no. 3, 251–268.MR1756260 (2001c:46123)

9. Izumi, M.,Finite group actions onC∗-algebras with the Rohlin property. II, Adv. Math.184(2004), no. 1, 119–160.MR2047851 (2005b:46153)

10. Katsura, T.,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras I, fundamental results, Trans. Amer. Math. Soc.356 (2004), no. 11, 4287–4322.MR2067120 (2005b:46119)

11. Katsura, T.,A construction ofC∗-algebras fromC∗-correspondences, Advances in Quan-tum Dynamics, Contemp. Math.335, Amer. Math. Soc., Providence, RI (2003), 173–182.MR2029622 (2005k:46131)

12. Katsura, T.,OnC∗-algebras associated withC∗-correspondences, J. Funct. Anal.217(2004),no. 2, 366–401.MR2102572 (2005e:46099)

13. Katsura, T.,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras IV, pure infiniteness, preprint 2005, math.OA/0509343.cf. MR 2008m:46143

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16. Kumjian, A., Pask, D., Raeburn, I.,Cuntz-Krieger algebras of directed graphs, Pac. J. Math.184(1998), no. 1, 161–174.MR1626528 (99i:46049)

17. Phillips, N. C.,A classification theorem for nuclear purely infinite simpleC∗-algebras, Doc.Math.5 (2000), 49–114.MR1745197 (2001d:46086b)

18. Pimsner, M. V.,A class ofC∗-algebras generalizing both Cuntz-Krieger algebras and crossedproducts byZ, Free probability theory, Fields Inst. Commun.12,Amer. Math. Soc., Providence,RI (1997), 189–212.MR1426840 (97k:46069)

19. Raeburn, I.,Graph algebras, CBMS Reg. Conf. Ser. Math.103,Published for the ConferenceBoard of the Mathematical Sciences, Washington, DC, by the American Mathematical Society,Providence, RI, 2005.MR2135030 (2005k:46141)

20. Rørdam, M.,Classification of nuclearC∗-algebras, Encyclop. Math. Sci.126,Operator Alge-bras and Non-commutative Geometry7, Springer-Verlag, Berlin (2002), 1–145.MR1878882(2003i:46060)

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22. Spielberg, J.,Non-cyclotomic Presentations of Modules and Prime-order Automorphisms ofKirchberg Algebras, preprint 2005, math.OA/0504287.cf. MR2377136

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2009

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Citations

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MR2386934 (2008m:46143)46L55 (19K56 37B10 46L05 46L80)

Katsura, Takeshi (J-HOKK)A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras.IV. Pure infiniteness. (English summary)J. Funct. Anal.254(2008),no. 5,1161–1187.

In this final installment of the series of papers concerningC∗-algebras associated with topologicalgraphs [Part III, T. Katsura, Ergodic Theory Dynam. Systems26 (2006), no. 6, 1805–1854;MR2279267 (2007j:46090)], the author gives a sufficient condition so that theseC∗-algebras aresimple and purely infinite. As a corollary, he constructs all Kirchberg algebras asC∗-algebrasassociated with topological graphs.

A topological graphE = (E0, E1, d, r) is viewed as a sort of dynamical system, and the asso-ciatedC∗-algebraO(E) as a sort of crossed product byN. In a previous paper in the series, theauthor extended many notions from dynamical systems such as orbits and minimality to topologi-cal graphs. A notion of contracting topological graphs is now introduced, and the first main resultis:

Theorem A. For a minimal and contracting topological graphE, theC∗-algebraO(E) is simpleand purely infinite.

An interesting question is if the converse of this theorem is true.The author also presents a new construction of all simple, separable, nuclear, purely infiniteC∗-

algebras satisfying the Universal Coefficient Theorem in order to attack some open problems. Thecorresponding topological graph is a sort of skew product of a discrete graph with the unit circle.His second main result is:

Theorem C. All Kirchberg algebras appear asC∗-algebras of topological graphs.Reviewed byValentin Deaconu

References

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11. T. Katsura, A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras I, fundamental results, Trans. Amer. Math. Soc. 356 (11) (2004) 4287–4322.MR2067120 (2005b:46119)

12. T. Katsura, A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras III, ideal structures, Ergodic Theory Dynam. Systems 26 (6) (2006) 1805–1854.MR2279267 (2007j:46090)

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14. T. Katsura, Topological graphs and singly generated dynamical systems, in preparation.

15. T. Katsura, A construction of actions on Kirchberg algebras which induce given actions ontheirK-groups, J. Reine Angew. Math., in press.MR2400990 (2009b:46140)

16. E. Kirchberg, The classification of purely infiniteC∗-algebras using Kasparov’s theory,preprint, 1994.

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25. J.S. Spielberg, Semiprojectivity for certain purely infiniteC∗-algebras, Trans. Amer. Math.Soc., in press.

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27. J. Spielberg, Non-cyclotomic presentations of modules and prime-order automorphisms ofKirchberg algebras, J. Reine Angew. Math., in press.

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2008, 2009

Article

Citations

From References: 1From Reviews: 0

MR2413377 (2009b:46118)46L05 (46L55)

Katsura, Takeshi (J-HOKK)Ideal structure of C∗-algebras associated withC∗-correspondences. (English summary)Pacific J. Math.230(2007),no. 1,107–145.

Given aC∗-correspondenceX (sometimes also called a Hilbert bimodule) over aC∗-algebraA,Pimsner defined a way to construct aC∗-algebraOX that is universal for certain representations ofthe pair(X, A). TheC∗-algebras of the formOX , now called Cuntz-Pimnser algebras, include avery large class ofC∗-algebras, and consequently a great deal of effort has been spent examininghow the structure ofOX is codified in the properties of the pair(X, A).

When defining theC∗-algebraOX , Pimsner required that the left action ofA onX be injective,and only gave a definition for this situation. In [J. Funct. Anal.217 (2004), no. 2, 366–401;MR2102572 (2005e:46099)] the author of the paper under review gave a generalized definitionof OX for a generalC∗-correspondenceX with no restriction on the left action. This generalizeddefinition deals with non-injective left actions in precisely the right way, and ensures that theassociated algebra is the appropriate algebra for various situations. In addition, it allows newclasses ofC∗-algebras to be included in the collection of “Cuntz-Pimsner algebras”, and thesenew classes are often natural as well as useful in applications.

In the paper under review the author classifies the gauge-invariant ideals of the (generalized)Cuntz-Pimsner algebras. In particular, the author shows that ifX is a C∗-correspondence overA, then the gauge-invariant ideals ofOX are in one-to-one correspondence with certain pairsof ideals inA, which the author callsO-pairs. To obtain this result, the author uses the gauge-invariant uniqueness theorem for Cuntz-Pimnser algebras, and shows that quotients of Cuntz-Pimsner algebras by gauge-invariant ideals are themselves Cuntz-Pimsner algebras. Throughoutthis analysis it is critical that the author is using his generalized definition of Cuntz-Pimsnersalgebras, because even whenOX is the Cuntz-Pimsner algebra of aC∗-correspondenceX withinjective left action, the quotient ofOX by a gauge-invariant ideal may be the Cuntz-Pimnseralgebra of aC∗-correspondence with non-injective left action. This gives further evidence of theimportance of the author’s generalized definition, and it shows that even if one is only interestedin Cuntz-Pimsner algebras ofC∗-correspondences with injective left action, the analysis of thesealgebras will still require one to consider Cuntz-Pimsner algebras ofC∗-correspondences in whichthe left action is not injective.

One important special case of Cuntz-Pimnser algebras, and also a class that motivated manyof the results in this paper, is that of graphC∗-algebras. IfE is a directed graph andC∗(E) isthe associatedC∗-algebra, then it is known that the gauge-invariant ideals ofC∗(E) correspondto admissible pairs(H,S) consisting of a saturated hereditary subset of verticesH and a subsetof breaking verticesS. The author’sO-pairs of ideals may be thought of as generalizations ofthese admissible pairs. Furthermore, when the graphE is row-finite with no sinks, theS sets arealways empty, and the gauge-invariant ideals ofC∗(E) are in one-to-one correspondence with the

saturated hereditary sets of vertices. Likewise, when aC∗-correspondenceX has a left action thatacts as compact operators and is injective, the second ideal in eachO-pair is zero, and the gauge-invariant ideals ofOX are in one-to-one correspondence with ideals ofA that are invariant withrespect toX (see Corollary 8.7 of the paper under review).

Reviewed byMark Tomforde

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2009

Article

Citations

From References: 5From Reviews: 1

MR2279267 (2007j:46090)46L05 (37B99 46L55)

Katsura, Takeshi (J-HOKK)A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras.III. Ideal structures. (English summary)Ergodic Theory Dynam. Systems26 (2006),no. 6,1805–1854.

In this third paper of this series, the author investigates the ideal structure ofC∗-algebras associ-ated with topological graphs. He gives a complete description of the gauge invariant ideals, andconditions for theseC∗-algebras to be simple, prime or primitive. He constructs a discrete graphfor which the associatedC∗-algebra is an inductive limit of finite-dimensionalC∗-algebras, and itis prime but not primitive.

A topological graphE = (E0, E1, d, r) consists of two locally compact spacesE0 (vertices)

andE1 (edges) and two continuous mapsd, r:E1→ E0 (domain and range), such thatd is alsoa local homeomorphism. The corresponding Cuntz-Pimsner algebraO(E) is defined using aC∗-correspondence overC0(E0), and a modification of the Pimsner construction. There is a naturalaction of the unit circle onO(E), called the gauge action. Topological graphs generalize dynamicalsystems and discrete graphs, and the analysis of the ideal structure ofO(E) is somewhat similarto the ideal structure of the corresponding crossed products or graph algebras. In his results, theauthor uses concepts from dynamical systems, adapted to topological graphs. These conceptsinclude orbits, periodic and aperiodic points, topological freeness, minimality and topologicaltransitivity. Some of his theorems were independently proved by Muhly and Tomforde in the moregeneral context of topological quivers.{For Part II see [T. Katsura, Internat. J. Math.17 (2006), no. 7, 791–833;MR2253144

(2007e:46051)].}Reviewed byValentin Deaconu

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2. T. Bates, D. Pask, I. Raeburn and W. Szymanski. TheC∗-algebras of row-finite graphs.NewYork J. Math.6 (2000), 307–324.MR1777234 (2001k:46084)

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7. T. Katsura. A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras II, examples.Internat. J. Math.17(7) (2006), 791–833.MR2253144 (2007e:46051)

8. T. Katsura. Ideal structure ofC∗-algebras associated withC∗-correspondences.Pacific J. Math.to appear.cf. MR 2005e:46099

9. T. Katsura. Non-separableAF-algebras.Operator Algebras: The Abel Symposium 2004 (AbelSymposia).Vol. 1. Springer, Berlin, 2006, pp. 165–174.MR2265049 (2007g:47117)

10. T. Katsura. Topological graphs and singly generated dynamical systems, in preparation.11. A. Kumjian, D. Pask, I. Raeburn and J. Renault. Graphs, groupoids, and Cuntz–Krieger alge-

bras.J. Funct. Anal.144(2) (1997), 505–541.MR1432596 (98g:46083)12. P. S. Muhly and M. Tomforde. Topological quivers.Internat. J. Math.16(7) (2005), 693–755.

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Theory48(3) (2002), 645–662.MR1962477 (2004h:46066)14. I. Raeburn and D. P. Williams.Morita Equivalence and Continuous-traceC∗-algebras (Mathe-

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c© Copyright American Mathematical Society 2007, 2009

Article

Citations

From References: 1From Reviews: 1

MR2265049 (2007g:47117)47L40Katsura, Takeshi (J-HOKK)Non-separable AF-algebras. (English summary)Operator Algebras: The Abel Symposium2004, 165–173,Abel Symp., 1,Springer, Berlin, 2006.

Commonly, an AFC∗-algebraA is defined to be aC∗-algebra for which there exists an increasingsequence of finite-dimensionalC∗-subalgebras whose union is dense inA. In the present paperthe author uses a generalization of this definition and refers to aC∗-algebraA as AF if it has adirected family of finite-dimensionalC∗-subalgebras whose union is dense inA. For separableC∗-algebras this latter definition coincides with the former, and therefore the focus of the presentpaper is on those AFC∗-algebras which are not separable.

The author shows that with this generalized definition it is possible to construct examples ofnon-separable AFC∗-algebras which exhibit properties not found in their separable counterparts.For example, it is known from the work of O. Bratteli [Trans. Amer. Math. Soc.171(1972), 195–

234; MR0312282 (47 #844)] that when the directed set in the above definition isZ+ (so thatthere is a sequence of subalgebras whose union is dense), the Bratteli diagram corresponding tothe sequence of finite-dimensional subalgebras determines the corresponding AF algebra up toisomorphism. The author gives an example, using as directed set the set of all finite subsets of afixed infinite set, of two AF algebras which, although not isomorphic, have isomorphic Brattelidiagrams. This example also serves to show that Elliott’s theorem for classifying separable AFalgebras usingK0 groups [G. A. Elliott, J. Algebra38 (1976), no. 1, 29–44;MR0397420 (53#1279)] does not follow for non-separable AF algebras.

In [N. Weaver, J. Funct. Anal.203 (2003), no. 2, 356–361;MR2003352 (2004g:46075)] it isshown that there existC∗-algebras which are prime but not primitive. The author of the presentpaper uses the generalized definition of AF algebra in order to construct an AF algebra with theseproperties, again using as directed set the set of all finite subsets of a fixed uncountable set.{For the entire collection seeMR2263583 (2007f:46002)}

Reviewed byRyan J. Zerr

c© Copyright American Mathematical Society 2007, 2009

Article

Citations

From References: 6From Reviews: 1

MR2253144 (2007e:46051)46L05 (37B99 46L55)

Katsura, Takeshi (J-HOKK)A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras.II. Examples. (English summary)Internat. J. Math.17 (2006),no. 7,791–833.

The author continues to studyC∗-algebras associated with topological graphs, and gives severalconstructions to produce new examples from old ones. These algebras are nuclear and satisfy theUCT, and contain almost all “classifiable”C∗-algebras.

Topological graphs generalize ordinary graphs and homeomorphisms on locally compact spaces.A topological graphE = (E0, E1, d, r) consists of two locally compact spacesE0 (vertices) andE1 (edges) and two continuous mapsd, r:E1→ E0 (domain and range), whered is also a localhomeomorphism. The corresponding Toeplitz algebraT(E) and Cuntz-Pimsner algebraO(E) aredefined using aC∗-correspondence overC0(E0), and a modification of the Pimsner construction.The author gives a new characterization of theseC∗-algebras in terms of their representationtheory and gauge actions.

A factor map between two topological graphs is defined to preserve the incidences and to havethe unique path lifting property, similarly to a covering map. It is shown that a factor map betweentwo such graphs induces *-homomorphisms between the associatedC∗-algebras.

The author also considers projective systems of graphs and their projective limits, and graphoperations which preserve the strongly Morita equivalence classes of theirC∗-algebras. A long

list of examples concludes the paper, including Exel-Laca algebras, Matsumoto algebras, crossedproducts by partial homeomorphisms, andC∗-algebras associated with branched coverings andwith singly generated dynamical systems.{For Part I see [T. Katsura, Trans. Amer. Math. Soc.356(2004), no. 11, 4287–4322 (electronic);

MR2067120 (2005b:46119)].}Reviewed byValentin Deaconu

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2. T. M. Carlsen and K. Matsumoto, Some remarks on theC∗-algebras associated with subshifts,Math. Scand.95(1) (2004) 145–160.MR2091486 (2005e:46093)

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19. K. Matsumoto,C∗-algebras associated with presentations of subshifts,Doc. Math.7 (2002),1–30.MR1911208 (2004j:46083)

20. P. S. Muhly and B. Solel, Tensor algebras, induced representations, and the Wold decomposi-tion, Canad. J. Math.51(4) (1999), 850–880.MR1701345 (2000i:46052)

21. M. V. Pimsner, A class ofC∗-algebras generalizing both Cuntz-Krieger algebras and crossedproducts byZ, in Free Probability Theory,Fields Institute Communication, Vol. 12 (AmericanMathematical Society, Providence, RI, 1997), pp. 189–212.MR1426840 (97k:46069)

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24. M. Rordam, A purely infiniteAH-algebra and an application toAF-embeddability,Israel J.Math.141(2004) 61–82.MR2063025 (2005c:46078)

25. M. Rørdam and E. Størmer,Classification of NuclearC∗-Algebras. Entropy in OperatorAlgebras,Encyclopaedia of Mathematical Sciences, Vol. 126; Operator Algebras and Non-commutative Geometry, Vol. 7 (Springer-Verlag, Berlin, 2002).MR1878881 (2002i:46047)

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2007, 2009

Article

Citations

From References: 2From Reviews: 0

MR2212029 (2006k:46088)46L05 (46L80 47B99)

Katsura, Takeshi (J-HOKK)C∗-algebras generated by scaling elements. (English summary)J. Operator Theory55 (2006),no. 1,213–222.

The theme of this paper is that scaling elements, a notion due to B. E. Blackadar and J. Cuntz [Amer.J. Math.104(1982), no. 4, 813–822;MR0667536 (84c:46057)], can be viewed as generalizationsof isometries. Several results on isometries (the Wold decomposition, Coburn’s theorem) are

generalized to scaling elements. It is shown thatB(H) has a large supply of scaling elements.Reviewed byDaniel Kucerovsky

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1. B.E. Blackadar, J. Cuntz, The structure of stable algebraically simpleC∗-algebras,Amer. J.Math.104(1982), 813–822.MR0667536 (84c:46057)

2. L.A. Coburn, TheC∗-algebra generated by an isometry,Bull. Amer. Math. Soc. (N.S.)73(1967),722–726.MR0213906 (35 #4760)

3. K.R. Davidson,C∗-Algebras by Example, Fields Inst. Monographs, vol. 6, Amer. Math. Soc.,Providence, RI 1996.MR1402012 (97i:46095)

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5. T. Katsura, AF-embeddability of crossed products of Cuntz algebras,J. Funct. Anal.196(2002),427–442.MR1943097 (2003i:46075)

6. T. Katsura, A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras. I. Fundamental results,Trans. Amer. Math. Soc.356(2004), 4287–4322.MR2067120(2005b:46119)

7. T. Katsura, OnC∗-algebras associated withC∗-correspondences,J. Funct. Anal.217(2004),366–401.MR2102572 (2005e:46099)

8. T. Katsura, A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras. II. Examples,Internat. J. Math., to appear.cf. MR 2007e:46051

9. M.V. Pimsner, A class ofC∗-algebras generalizing both Cuntz-Krieger algebras and crossedproducts byZ, in Free Probability Theory, Fields Inst. Commun., vol. 12, Amer. Math. Soc.,Providence, RI, 1997, pp. 189–212.MR1426840 (97k:46069)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2006, 2009

Article

Citations

From References: 20From Reviews: 1

MR2102572 (2005e:46099)46L05 (46L55)

Katsura, Takeshi (J-TOKYO)On C∗-algebras associated withC∗-correspondences. (English summary)J. Funct. Anal.217(2004),no. 2,366–401.

M. V. Pimsner [inFree probability theory (Waterloo, ON, 1995), 189–212, Amer. Math. Soc.,Providence, RI, 1997;MR1426840 (97k:46069)] introduced a way to construct aC∗-algebraOX

from a pair(A,X), whereA is aC∗-algebra andX is aC∗-correspondence (sometimes called a

Hilbert bimodule) overA. Throughout his analysis Pimsner assumed that his correspondence wasfull and that the left action ofA onX was injective. These Cuntz-Pimsner algebras have been foundto compose a class ofC∗-algebras that is extraordinarily rich and includes numerousC∗-algebrasfound in the literature: crossed products by automorphisms, crossed products by endomorphisms,partial crossed products, Cuntz-Krieger algebras,C∗-algebras of graphs with no sinks, Exel-Lacaalgebras, and many more.

Although in his initial work Pimsner assumed that hisC∗-correspondences were full and hadinjective left action, in recent years there have been efforts to remove these restrictions. Pimsnerhimself [op. cit. (Remark 1.2(3))] described how to deal with the case whenX is not full, definingthe so-called augmented Cuntz-Pimsner algebras. However, the case when the left action is notinjective has been more elusive. In [N. J. Fowler, P. S. Muhly and I. Raeburn, Indiana Univ. Math.J.52 (2003), no. 3, 569–605;MR1986889 (2005d:46114)(Proposition 1.3)] it was shown that foranyC∗-correspondenceX and for any idealK of A consisting of elements that act as compactoperators on the left ofX, one may defineO(K, X) to be aC∗-algebra which satisfies a certainuniversal property. In the case thatX is full with injective left action, this definition agrees withpreviously defined notions of relative Cuntz-Pimsner algebras, and the Cuntz-Pimsner algebraOX

is equal toO(J(X), X), whereJ(X) denotes the ideal consisting of all elements ofA which acton the left ofX as compact operators.

In [N. J. Fowler, P. S. Muhly and I. Raeburn, op. cit.] it was proposed that for a generalC∗-correspondenceX, the C∗-algebraO(J(X), X) is the proper analogue of the Cuntz-Pimsneralgebra. However, upon further analysis it seems that this is not exactly correct. To see why,consider the case of graphC∗-algebras. IfE = (E0, E1, r, s) is a graph, then there is a naturalC∗-correspondenceX(E) overC0(E0) associated toE [see N. J. Fowler and I. Raeburn, IndianaUniv. Math. J.48 (1999), no. 1, 155–181;MR1722197 (2001b:46093)(Example 1.2)]. IfE hasno sinks, then theC∗-algebraO(J(X(E)), X(E)) is isomorphic to the graphC∗-algebraC∗(E).However, whenE has sinks this will not necessarily be the case.

It is worth mentioning that graphs with sinks play an important role in the study of graphC∗-algebras. Even if one begins with a graphE containing no sinks, an analysis ofC∗(E) will oftennecessitate consideringC∗-algebras of graphs with sinks. For example, quotients ofC∗(E) willoften be isomorphic toC∗-algebras of graphs with sinks even whenE has no sinks. Consequently,one needs a theory that incorporates these objects.

This deficiency in the generalization of Cuntz-Pimsner algebras was addressed by Katsura firstin [Advances in quantum dynamics (South Hadley, MA, 2002), 173–182, Contemp. Math., 335,Amer. Math. Soc., Providence, RI, 2003;MR2029622 (2005k:46131)] and now in the paperunder review. IfX is aC∗-correspondence over aC∗-algebraA with left actionϕ:A→ L(X),then Katsura has proposed that the appropriate analogue of the Cuntz-Pimsner algebra isOX :=O(JX , X), where

JX := {a ∈ J(X): ab = 0 ∀ b ∈ ker ϕ}.(Note that whenϕ is injectiveJX = J(X).) It turns out that whenϕ is injective,OX is equal

to the augmented Cuntz-Pimsner algebra ofX, and whenX is also full,OX coincides with theCuntz-Pimsner algebra ofX. Furthermore, ifE is a graph (possibly containing sinks), thenOX(E)

is isomorphic toC∗(E).Additionally, there are many other reasons to use this definition ofOX for a generalC∗-

correspondenceX. For example, using Katsura’s definition, the following four facts are true:1. The class ofC∗-algebras associated toC∗-correspondences includes allC∗-algebras in sev-

eral interesting classes; e.g., all graphC∗-algebras, allC∗-algebras of topological graphs [T.Katsura, Trans. Amer. Math. Soc.356 (2004), no. 11, 4287–4322 (electronic);MR2067120(2005b:46119)], all C∗-algebras of topological quivers [P. S. Muhly and M. Tomforde, “Topo-logical quivers”, preprint, arxiv.org/abs/math/0312109], and crossed products by HilbertC∗-modules [B. Abadie, S. Eilers and R. Exel, Trans. Amer. Math. Soc.350 (1998), no. 8,3043–3054;MR1467459 (98k:46109)].

2. The class ofC∗-algebras associated toC∗-correspondences is closed under quotients bygauge-invariant ideals, and any gauge-invariant ideal ofOX is isomorphic to theC∗-algebraof someC∗-correspondence (both results are proven in [T. Katsura, “Ideal structure ofC∗-algebras associated withC∗-correspondences”, preprint, arxiv.org/abs/math/0309294]).These are facts that are important in the analysis of the ideal structure of theOX .

3. Constructions forC∗-algebras of graphs with sinks may be generalized to theOX ’s (e.g.adding tails to sinks [P. S. Muhly and M. Tomforde, Doc. Math.9 (2004), 79–106 (electronic);MR2054981 (2005a:46117)]).

4. TheC∗-algebraOX is the smallestC∗-algebra generated by injective representations ofXadmitting gauge actions [T. Katsura, op. cit., preprint (Proposition 7.14)].

These facts, among others, provide strong evidence that Katsura has given the “correct definition”of theC∗-algebra associated to aC∗-correspondence, and ensure theOX ’s created in this way haveproperties that are reasonable, natural, and useful in their analysis.

In the paper under review the author proves versions of the gauge-invariant uniqueness theoremfor OX and also gives conditions onX for OX to be nuclear, exact, or satisfy the universalcoefficients theorem. In addition, the author obtains a cyclic 6-term exact sequence ofK-groupsthat in certain situations is useful for computing theK-theory ofOX .

Reviewed byMark Tomforde

References

1. B. Abadie, S. Eilers, R. Exel, Morita equivalence for crossed products by HilbertC∗-bimodules,Trans. Amer. Math. Soc. 350 (8) (1998) 3043–3054.MR1467459 (98k:46109)

2. T. Bates, J. Hong, I. Raeburn, W. Szymanski, The ideal structure of theC∗-algebras of infinitegraphs, Illinois J. Math. 46 (4) (2002) 1159–1176.MR1988256 (2004i:46105)

3. L.G. Brown, Stable isomorphism of hereditary subalgebras ofC∗-algebras, Pacific J. Math. 71(2) (1977) 335–348.MR0454645 (56 #12894)

4. S. Doplicher, R. Longo, J.E. Roberts, L. Zsido, A remark on quantum group actions andnuclearity, Rev. Math. Phys. 14 (7–8) (2002) 787–796.MR1932666 (2003h:46097)

5. K. Dykema, D. Shlyakhtenko, Exactness of Cuntz–PimsnerC∗-algebras, Proc. EdinburghMath. Soc. 44 (2001) 425–444.MR1880402 (2003a:46084)

6. R. Exel, A Fredholm operator approach to Morita equivalence,K-Theory 7 (3) (1993) 285–308.

MR1244004 (94h:46107)7. N.J. Fowler, M. Laca, I. Raeburn, TheC∗-algebras of infinite graphs, Proc. Amer. Math. Soc.

128 (8) (2000) 2319–2327.MR1670363 (2000k:46079)8. N.J. Fowler, P.S. Muhly, I. Raeburn, Representations of Cuntz–Pimsner algebras, Indiana Univ.

Math. J. 52 (3) (2003) 569–605.MR1986889 (2005d:46114)9. N. Higson, J. Roe, AnalyticK-homology, Oxford Mathematical Monographs, Oxford Science

Publications, Oxford University Press, Oxford, 2000.MR1817560 (2002c:58036)10. T. Kajiwara, C. Pinzari, Y. Watatani, Ideal structure and simplicity of theC∗-algebras generated

by Hilbert bimodules, J. Funct. Anal. 159 (2) (1998) 295–322.MR1658088 (2000a:46094)11. T. Katsura, A class ofC∗-algebras generalizing both graph algebras and homeomorphism

C∗-algebras I, fundamental results, Trans. Amer. Math. Soc., to appear.cf. MR 2005b:4611912. T. Katsura, A construction ofC∗-algebras fromC∗-correspondences, in: G.L. Price, B.M.

Baker, P.E.T. Jorgensen, P.S. Muhly (Eds.), Advances in Quantum Dynamics, Contemp. Math.,vol. 335, American Mathematical Society, Providence, RI, 2003, pp. 173–182.MR2029622(2005k:46131)

13. T. Katsura, Ideal structure ofC∗-algebras associated withC∗-correspondences, Preprint 2003,math.OA/0309294.cf. MR 2005e:46099

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16. A. Kumjian, D. Pask, I. Raeburn, J. Renault, Graphs, groupoids, and Cuntz–Krieger algebras,J. Funct. Anal. 144 (2) (1997) 505–541.MR1432596 (98g:46083)

17. C. Lance, On nuclearC∗-algebras, J. Funct. Anal. 12 (1973) 157–176.MR0344901 (49 #9640)18. E.C. Lance, HilbertC∗-modules, A Toolkit for Operator Algebraists, London Mathemati-

cal Society Lecture Note Series, Vol. 210, Cambridge University Press, Cambridge, 1995.MR1325694 (96k:46100)

19. P.S. Muhly, B. Solel, Tensor algebras overC∗-correspondences: representations, dilations, andC∗-envelopes, J. Funct. Anal. 158 (2) (1998) 389–457.MR1648483 (99j:46066)

20. M.V. Pimsner, A class ofC∗-algebras generalizing both Cuntz–Krieger algebras and crossedproducts byZ, in: D. Voiculescu (Ed.), Free Probability Theory, Fields Inst. Communication,Vol. 12, American Mathematical Society, Providence, RI, 1997, pp. 189–212.MR1426840(97k:46069)

21. J. Rosenberg, C. Schochet, The Kunneth theorem and the universal coefficient theorem for Kas-parov’s generalizedK-functor, Duke Math. J. 55 (2) (1987) 431–474.MR0894590 (88i:46091)

22. S. Wassermann, ExactC∗-algebras and related topics, Lecture Notes Series, Vol. 19, SeoulNational University, Research Institute of Mathematics, Global Analysis Research Center,Seoul, 1994.MR1271145 (95b:46081)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2005, 2009

Article

Citations

From References: 24From Reviews: 5

MR2067120 (2005b:46119)46L05 (37B99 46L55 46L80)

Katsura, Takeshi (J-TOKYO)A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras.I. Fundamental results. (English summary)Trans. Amer. Math. Soc.356(2004),no. 11,4287–4322 (electronic).

The class ofC∗-algebras under consideration is constructed using representations ofC∗-correspondences, following fundamental ideas of M. Pimsner. The author uses the topologicalgraphs to unify the usual graphC∗-algebras of A. Kumjian, I. Raeburn, J. Renault and others, thehomeomorphismC∗-algebras studied by J. Tomiyama, and the generalized Cuntz-Krieger alge-bras of V. Deaconu. A key ingredient is the notion of relative Cuntz-Pimsner algebra introducedby P. Muhly and B. Solel.

The main results of the paper are the gauge-invariant uniqueness theorem, the Cuntz-Kriegeruniqueness theorem, the fact that the author’sC∗-algebras are nuclear and satisfy the UCT, andthe existence of a six-term exact sequence ofK-theory.

A topological correspondence between locally compact spacesE0 andF 0 is a triple(E1, d, r),whereE1 is a locally compact space,d:E1→ E0 is a local homeomorphism, andr:E1→ F 0 isany continuous map. This defines a HilbertC0(E0)-moduleCd(E1) by

Cd(E1) = {ξ ∈ C(E1): 〈ξ, ξ〉 ∈ C0(E0)},where

〈ξ, ξ〉(v) =∑

e∈d−1(v)

|ξ(e)|2, v ∈ E0

andξf(e) = ξ(e)f(d(e)), ξ ∈ Cd(E1), f ∈ C0(E0).

The left action is given by

πr:C0(F 0)→ L(Cd(E1)), (πr(f)ξ)(e) = f(r(e))ξ(e).

WhenF 0 = E0, we get a topological graph, withE0 the set of vertices, andE1 the set of edges.In that case,Cd(E1) becomes a Hilbert bimodule.

A ToeplitzE-pair on aC∗-algebraA is a pair of mapsT = (T 0, T 1), whereT 0:C0(E0)→A isa *-homomorphism, andT 1:Cd(E1)→A is a linear map satisfying

(i) T 1(ξ)∗T 1(η) = T 0(〈ξ, η〉) for ξ, η ∈ Cd(E1),(ii) T 0(f)T 1(ξ) = T 1(πr(f)ξ) for f ∈ C0(E0) andξ ∈ Cd(E1).By definition,T(E) denotes the universalC∗-algebra generated by a ToeplitzE-pair. A Toeplitz

E-pair is called Cuntz-Krieger ifT 0(f) = Φ1(πr(f)) for all f ∈ C0(E0rg), whereΦ1(θξ,η) =

T 1(ξ)T 1(η)∗, for θξ,η a rank one operator, andE0rg is a certain subset (regular vertices) ofE0.

O(E) denotes the universalC∗-algebra generated by a Cuntz-KriegerE-pair.WhenE0 is discrete,O(E) is a graph algebra. IfE0 = X is a compact space,d = id, andr =

σ is a homeomorphism ofX, thenO(E) is the crossed productC(X)×α Z, whereα(f)(x) =f(σ−1x). The general case provides a very rich and tractable class ofC∗-algebras.

There is a short exact sequence0 → IE → T(E) → O(E) → 0 which gives a six-term ex-act sequence ofK-theory. SinceIE andT(E) areKK-equivalent withC0(E0

rg) andC0(E0),respectively, the author obtains the exact sequence

K0(C0(E0rg))→K0(C0(E0))→K0(O(E))

↑ ↓

K1(O(E))←K1(C0(E0))←K1(C0(E0rg)).

He uses this to give a new proof of the computation ofK-groups of graph algebras.Reviewed byValentin Deaconu

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4. Bates, T.; Pask, D.; Raeburn, I.; Szyma’nski, W.The C∗-algebras of row-finite graphs.NewYork J. Math.6 (2000), 307–324.MR1777234 (2001k:46084)

5. Cuntz, J.; Krieger, W.A class of C∗-algebras and topological Markov chains.Invent. Math.56(1980), no. 3, 251–268.MR0561974 (82f:46073a)

6. Deaconu, V.Continuous graphs and C*-algebras.Operator theoretical methods, 137– 149,Theta Found., Bucharest, 2000.MR1770320 (2001g:46123)

7. Deaconu, V.; Kumjian, A.; Muhly, P.Cohomology of topological graphs and Cuntz- Pimsneralgebras.J. Operator Theory46 (2001), no. 2, 251–264.MR1870406 (2003a:46093)

8. Drinen, D.; Tomforde, M.The C∗-algebras of arbitrary graphs.To appear in Rocky MountainJ. Math.cf. MR 2006h:46051

9. Drinen, D.; Tomforde, M.Computing K-theory andExt for graph C∗-algebras.Illinois J. Math.46 (2002), no. 1, 81–91.MR1936076 (2003k:46103)

10. Dykema, K.; Shlyakhtenko, D.Exactness of Cuntz-Pimsner C*-algebras.Proc. Edin- burghMath. Soc.44 (2001), 425-444.MR1880402 (2003a:46084)

11. Exel, R.A Fredholm operator approach to Morita equivalence. K-Theory 7 (1993), no. 3,285–308.MR1244004 (94h:46107)

12. Exel, R.; Laca, M.Cuntz-Krieger algebras for infinite matrices.J. Reine Angew. Math.512(1999), 119–172.MR1703078 (2000i:46064)

13. Exel, R.; Laca, M.; Quigg, J.Partial dynamical systems and C∗-algebras generated by partialisometries.J. Operator Theory47 (2002), no. 1, 169–186.MR1905819 (2003f:46108)

14. Fowler, N. J.; Laca, M.; Raeburn, I.The C∗-algebras of infinite graphs.Proc. Amer. Math. Soc.

128(2000), no. 8, 2319–2327.MR1670363 (2000k:46079)15. Fowler, N. J.; Raeburn, I.The Toeplitz algebra of a Hilbert bimodule.Indiana Univ. Math. J.

48 (1999), no. 1, 155–181.MR1722197 (2001b:46093)16. Hong, J. H.; Szymanski, W.The primitive ideal space of the C∗-algebras of infinite graphs.J.

Math. Soc. Japan56 (2004), no. 1, 45–64.MR2023453 (2004j:46088)17. Kajiwara, T.; Pinzari, C.; Watatani, Y.Ideal structure and simplicity of the C∗- algebras

generated by Hilbert bimodules.J. Funct. Anal.159 (1998), no. 2, 295–322.MR1658088(2000a:46094)

18. Katsura, T.The ideal structures of crossed products of Cuntz algebras by quasi-free actions ofabelian groups.Canad. J. Math.55 (2003), no. 6, 1302–1338.MR2016248 (2004j:46090)

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21. Katsura, T.A class of C∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras IV, pure infiniteness.In preparation.

22. Katsura, T. On C∗-algebras associated with C∗-correspondences.Preprint 2003,math.OA/0309088, to appear in J. Funct. Anal.cf. MR 2005e:46099

23. Kumjian, A.; Pask, D.; Raeburn, I.Cuntz-Krieger algebras of directed graphs.Pacific J. Math.184(1998), no. 1, 161–174.MR1626528 (99i:46049)

24. Kumjian, A.; Pask, D.; Raeburn, I.; Renault, J.Graphs, groupoids, and Cuntz-Krieger algebras.J. Funct. Anal.144(1997), no. 2, 505–541.MR1432596 (98g:46083)

25. Lance, E. C.Hilbert C∗-modules. A toolkit for operator algebraists.London MathematicalSociety Lecture Note Series,210. Cambridge University Press, Cambridge, 1995.MR1325694(96k:46100)

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27. Pimsner, M. V.A class of C∗-algebras generalizing both Cuntz-Krieger algebras and crossedproducts by Z.Free probability theory, 189–212, Fields Inst. Commun.,12, Amer. Math. Soc.,Providence, RI, 1997.MR1426840 (97k:46069)

28. Raeburn, I.; Szyma’nski, W.Cuntz-Krieger algebras of infinite graphs and matrices.Preprint.cf. MR 2004i:46087

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30. Schweizer, J.Crossed products by C∗-correspondences and Cuntz-Pimsner algebras. C∗- al-gebras, 203–226, Springer, Berlin, 2000.MR1798598 (2002f:46133)

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Note: This list, extracted from the PDF form of the original paper, may contain data conversionerrors, almost all limited to the mathematical expressions.

c© Copyright American Mathematical Society 2005, 2009

Article

Citations

From References: 15From Reviews: 1

MR2029622 (2005k:46131)46L05 (46L55)

Katsura, Takeshi (J-TOKYO)A construction of C∗-algebras fromC∗-correspondences. (English summary)Advances in quantum dynamics(South Hadley, MA, 2002), 173–182,Contemp.Math., 335,Amer.Math.Soc., Providence, RI, 2003.

Since the Cuntz-Krieger algebras were introduced by J. Cuntz and W. Krieger [Invent. Math.56(1980), no. 3, 251–268;MR0561974 (82f:46073a); J. Cuntz, Invent. Math.63 (1981), no. 1, 25–40;MR0608527 (82f:46073b)], a number of generalizations of have been proposed. Examples ofsuch generalizations are some crossed products by partial actions [R. Exel and M. Laca, J. ReineAngew. Math.512(1999), 119–172;MR1703078 (2000i:46064)], as well as certainC∗-algebrasassociated to directed or topological graphs (for instance [V. Deaconu, Pacific J. Math.190(1999),no. 2, 247–260;MR1722892 (2000j:46103); in Operator theoretical methods (Timisoara, 1998),137–149, Theta Found., Bucharest, 2000;MR1770320 (2001g:46123); N. J. Fowler, M. Laca andI. Raeburn, Proc. Amer. Math. Soc.128(2000), no. 8, 2319–2327;MR1670363 (2000k:46079); A.Kumjian et al., J. Funct. Anal.144(1997), no. 2, 505–541;MR1432596 (98g:46083); A. Kumjian,D. Pask and I. Raeburn, Pacific J. Math.184(1998), no. 1, 161–174;MR1626528 (99i:46049)]).

Another important example is the so called Cuntz-Pimsner algebra: this is theC∗-algebra thatM. V. Pimsner [inFree probability theory (Waterloo, ON, 1995), 189–212, Amer. Math. Soc.,Providence, RI, 1997;MR1426840 (97k:46069)] associated to aC∗-correspondence. Recall thataC∗-correspondenceE over aC∗-algebraA is a bimodule overA, which is a HilbertA-moduleas a right module, and where the left action ofA is given by adjointable operators. Pimsner’sconstruction generalizes both Cuntz-Krieger algebras and the crossed product by an automor-phism. It gradually turned out that many of the generalizations of Cuntz-Krieger algebras werespecial cases of Cuntz-Pimsner algebras, or at least of relative Cuntz-Pimsner algebras [see P.

S. Muhly and B. Solel, J. Funct. Anal.158 (1998), no. 2, 389–457;MR1648483 (99j:46066);N. J. Fowler, P. S. Muhly and I. Raeburn, Indiana Univ. Math. J.52 (2003), no. 3, 569–605;MR1986889 (2005d:46114); J. Schweizer, inC∗-algebras (Munster, 1999), 203–226, Springer,Berlin, 2000;MR1798598 (2002f:46133); J. Funct. Anal.180(2001), no. 2, 404–425;MR1814994(2002f:46113)]. A construction closely related with the Cuntz-Pimnser algebra is the crossed prod-uct by a HilbertC∗-bimodule introduced in [B. Abadie, S. Eilers and R. Exel, Trans. Amer. Math.Soc.350(1998), no. 8, 3043–3054;MR1467459 (98k:46109)]. Although a HilbertC∗-bimoduleis a particular case of aC∗-correspondence, the corresponding crossed product and Cuntz-Pimsneralgebras do not coincide in general (in fact the crossed product agrees with the augmented Cuntz-Pimsner algebra).

In the paper under review, the author finds a common generalization (already expected in [B.Abadie, S. Eilers and R Exel, op. cit.]) of the Cuntz-Pimsner algebras and the crossed prod-ucts by HilbertC∗-bimodules. He considers an arbitraryC∗-correspondenceE. For such aC∗-correspondence he defines aC∗-algebraOE, which is a slight modification of Pimsner’s construc-tion. These two constructions agree whenE is both faithful (as a left module) and full (as a rightmodule). Moreover, there is a natural map fromE into OE which is always injective, so it ispossible to recover all the information aboutE from OE.

The paper is rather short. After giving the necessary definitions, the author introduces theC∗-algebraOE. In the rest of the paper the author shows that the following classes of algebras areincluded in the class defined by him: Cuntz-Pimsner algebras, relative Cuntz-Pimsner algebras,crossed products by HilbertC∗-bimodules,C∗-algebras of arbitrary graphs,C∗-algebras arisingfrom topological graphs, and crossed products by partial morphisms.{For the entire collection seeMR2023163 (2004i:00011)}

Reviewed byFernando Abadie

c© Copyright American Mathematical Society 2005, 2009

Article

Citations

From References: 4From Reviews: 0

MR2018231 (2004k:46121)46L55Katsura, Takeshi (J-TOKYO)On crossed products of the Cuntz algebraO∞ by quasi-free actions of abelian groups.(English summary)Operator algebras and mathematical physics(Constanta, 2001), 209–233,Theta, Bucharest,2003.

In the paper under review, the author deals with crossed products of the CuntzC∗-algebraO∞by quasi-free actions of abelian, locally compact groupsG (such actions rescale the standardgenerators ofO∞). This is a continuation and a natural extension of the same author’s previouswork on similar actions on Cuntz algebrasOn [Canad. J. Math.55 (2003), no. 6, 1302–1338;

MR2016248 (2004j:46090)]. The main result of the present paper is a complete description of theideal structure ofO∞o G. In addition, the strong Connes spectrum for the actions in questionis calculated. The analysis of the ideal structure ofO∞ o G bears an interesting resemblenceto the work on ideals of theC∗-algebras of infinite graphs [J. H. Hong and W. Szymanski, J.Math. Soc. Japan56 (2004), no. 1, 45–64;MR2023453 (2004j:46088); T. Bates et al., Illinois J.Math. 46 (2002), no. 4, 1159–1176;MR1988256 (2004i:46105)]. In fact, the crossed productsconsidered in the paper under review may be viewed as an example of a more general constructionof Cuntz-Krieger algebras corresponding to continuous analogues of directed graphs, subsequentlydeveloped by the author [Surikaisekikenkyusho Kokyuroku No. 1291 (2002), 73–83MR1983631].

Reviewed byWojciech Szymanski

c© Copyright American Mathematical Society 2004, 2009

Article

Citations

From References: 2From Reviews: 1

MR2016248 (2004j:46090)46L55 (46L05 46L45 46L80)

Katsura, Takeshi (J-TOKYO)The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abeliangroups. (English summary)Canad. J. Math.55 (2003),no. 6,1302–1338.

In this paper Katsura determines the ideal structure of crossed products of Cuntz algebras underquasi-free actions of abelian groups. The primitive ideal structure of theseC∗-algebras is deter-mined, and the hull-kernel topology on this space described. A condition is given, analogous tocondition (II) of J. Cuntz [see Invent. Math.63 (1981), no. 1, 25–40;MR0608527 (82f:46073b)]and condition (K) for graphC∗-algebras [see, e.g., A. Kumjian, D. Pask and I. Raeburn, PacificJ. Math.184(1998), no. 1, 161–174;MR1626528 (99i:46049)], under which all ideals are invari-ant under an action ofT. Katsura gives necessary and sufficient conditions for the algebras to beprimitive, and computes the Connes spectra andK-groups of the algebras. The work follows theprogramme established in [A. an Huef and I. Raeburn, Ergodic Theory Dynam. Systems17(1997),no. 3, 611–624;MR1452183 (98k:46098)] for determining the ideal structure of Cuntz-Kriegeralgebras.

Reviewed byTeresa Gay Bates

References

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3. J. Cuntz,A class ofC∗-algebras and topological Markov chains. II. Reducible chains and theExt-functor forC∗-algebras.Invent. Math. (1)63(1981), 25–40.MR0608527 (82f:46073b)

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9. T. Katsura,AF-embeddability of crossed products of Cuntz algebras.J. Funct. Anal.196(2002),427–442.MR1943097 (2003i:46075)

10. T. Katsura,On crossed products of the Cuntz algebraO∞ by quasi-free actions of abeliangroups.Proceedings of the Operator Algebras and Mathematical Physics Conference, Con-stanta 2001, to appear.cf. MR 2004k:46121

11. T. Katsura,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras I, fundamental results.preprint.cf. MR 2005b:46119

12. T. Katsura,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras II, examples.preprint.cf. MR 2005b:46119

13. T. Katsura,A class ofC∗-algebras generalizing both graph algebras and homeomorphismC∗-algebras III, ideal structures.in preparation.

14. A. Kishimoto,Simple crossed products ofC∗-algebras by locally compact abelian groups.Yokohama Math. J. (1–2)28(1980), 69–85.MR0623751 (82g:46110)

15. A. Kishimoto and A. Kumjian,Simple stably projectionlessC∗-algebras arising as crossedproducts.Canad. J. Math. (5)48(1996), 980–996.MR1414067 (98b:46072)

16. A. Kishimoto and A. Kumjian,Crossed products of Cuntz algebras by quasi-free automor-phisms.Operator algebras and their applications, 173–192, Fields Inst. Commun.13, Amer.Math. Soc., Providence, RI, 1997.MR1424962 (98h:46076)

17. A. Kumjian and D. Pask,C∗-algebras of directed graphs and group actions.Ergodic TheoryDynam. Systems (6)19(1999), 1503–1519.MR1738948 (2000m:46125)

18. A. Kumjian, D. Pask, I. Raeburn and J. Renault,Graphs, groupoids, and Cuntz-Krieger alge-bras.J. Funct. Anal. (2)144(1997), 505–541.MR1432596 (98g:46083)

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Providence, RI, 1997.MR1426840 (97k:46069)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR1980919 46L05 (46L55 46L80)

Katsura, Takeshi (J-TOKYO)A summary of the works “A class ofC∗-algebras generalizing both graph algebras andhomeomorphismC∗-algebras”.The structure of operator algebras and its applications (Japanese) (Kyoto, 2002).Surikaisekikenkyusho KokyurokuNo. 1300(2003), 88–101.

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MR1983631 46L05 (46L55)

Katsura, Takeshi (J-TOKYO)Continuous graphs and crossed products of Cuntz algebras.Recent aspects ofC∗-algebras (Japanese) (Kyoto, 2002).Surikaisekikenkyusho KokyurokuNo. 1291(2002), 73–83.

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From References: 3From Reviews: 0

MR1943097 (2003i:46075)46L55 (46L05)

Katsura, Takeshi (J-TOKYO)AF-embeddability of crossed products of Cuntz algebras. (English summary)J. Funct. Anal.196(2002),no. 2,427–442.

The author considers crossed products of the Cuntz algebraOn, the universalC∗-algebra generatedby n isometriesS1, S2, . . . , Sn with

∑ni=1 SiS

∗i = 1, by certain quasi-free actions of a locally

compact abelian groupG. He thus obtains examples of crossed products of purely infiniteC∗-algebras which are AF-embeddable.

Let Γ be the dual group ofG and fix ω = (ω1, ω2, . . . , ωn) ∈ Γn. Define an actionαω:G→AutOn by αω

t (Si) = ωi(t)Si. Fork ∈N, letW(k)n = {(i1, i2, . . . , ik): ij ∈ {1, 2, . . . n}} be the set

of k-tuples, andWn =∞⋃

k=0

W(k)n . For eachµ = (i1, i2, . . . , ik) ∈Wn define an elementωµ of Γ by

ωµ =∑k

j=1 ωij . The closed semigroup generated byω is {ωµ:µ ∈Wn}.The main theorem of this paper is thatOn ×αω G is AF-embeddable provided−ωi /∈{ωµ:µ ∈Wn} for i = 1, 2, . . . , n. The author also generalises results of A. Kishimoto [Comm.Math. Phys.81(1981), no. 3, 429–435;MR0634163 (83c:46061)] and Kishimoto and A. Kumjian[in Operator algebras and their applications (Waterloo, ON, 1994/1995), 173–192, Amer. Math.Soc., Providence, RI, 1997;MR1424962 (98h:46076)] to conclude thatOn×αω G is simple andpurely infinite if and only if the closed semigroup generated byω is all of Γ. A dichotomy arises:if On ×αω G is simple then it is either AF-embeddable or purely infinite. In the final sectionAF-embeddability ofO∞×αω G is investigated.

Reviewed byAstrid An Huef

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abelian groups, preprint.cf. MR 2004j:460909. A. Kishimoto, Simple crossed products ofC∗-algebras by locally compact abelian groups,

Yokohama Math. J. 28 (1-2) (1980) 69–85.MR0623751 (82g:46110)10. A. Kishimoto, A. Kumjian, Simple stably projectionlessC∗-algebras arising as crossed prod-

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Operator algebras and their applications, Fields Institute Communications, Vol. 13, AmericanMathematical Society, Providence, RI, 1997, pp. 173–192.MR1424962 (98h:46076)

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2003, 2009

Article

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MR1915431 46L55 (46L05)

Katsura, Takeshi (J-TOKYO)Crossed products of Cuntz algebras by quasi-free actions of abelian groups.Theory of operator algebras and its applications (Japanese) (Kyoto, 2001).Surikaisekikenkyusho KokyurokuNo. 1250(2002), 9–15.

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