## Nonself-adjoint $2$-graph algebras

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- by Adam H. Fuller and Dilian Yang PDF
- Trans. Amer. Math. Soc.
**367**(2015), 6199-6224 Request permission

## Abstract:

We study the structure of weakly-closed nonself-adjoint algebras arising from representations of single vertex $2$-graphs. These are the algebras generated by $2$ isometric tuples which satisfy a certain commutation relation. We show that these algebras have a lower-triangular $3\times 3$ form. The left-hand side of this matrix decomposition is a slice of the enveloping von Neumann algebra generated by the $2$-graph algebra. We further give necessary and sufficient conditions for these algebras themselves to be von Neumann algebras. The paper concludes with further study of atomic representations.## References

- Kenneth R. Davidson,
*Free semigroup algebras. A survey*, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 209–240. MR**1882697** - Kenneth R. Davidson,
*$\scr B(\scr H)$ is a free semigroup algebra*, Proc. Amer. Math. Soc.**134**(2006), no. 6, 1753–1757. MR**2204288**, DOI 10.1090/S0002-9939-05-08147-5 - Kenneth R. Davidson, Elias Katsoulis, and David R. Pitts,
*The structure of free semigroup algebras*, J. Reine Angew. Math.**533**(2001), 99–125. MR**1823866**, DOI 10.1515/crll.2001.028 - Kenneth R. Davidson, David W. Kribs, and Miron E. Shpigel,
*Isometric dilations of non-commuting finite rank $n$-tuples*, Canad. J. Math.**53**(2001), no. 3, 506–545. MR**1827819**, DOI 10.4153/CJM-2001-022-0 - Kenneth R. Davidson, Jiankui Li, and David R. Pitts,
*Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras*, J. Funct. Anal.**224**(2005), no. 1, 160–191. MR**2139108**, DOI 10.1016/j.jfa.2004.08.005 - Kenneth R. Davidson and David R. Pitts,
*Invariant subspaces and hyper-reflexivity for free semigroup algebras*, Proc. London Math. Soc. (3)**78**(1999), no. 2, 401–430. MR**1665248**, DOI 10.1112/S002461159900180X - Kenneth R. Davidson, Stephen C. Power, and Dilian Yang,
*Atomic representations of rank 2 graph algebras*, J. Funct. Anal.**255**(2008), no. 4, 819–853. MR**2433954**, DOI 10.1016/j.jfa.2008.05.008 - Kenneth R. Davidson, Stephen C. Power, and Dilian Yang,
*Dilation theory for rank 2 graph algebras*, J. Operator Theory**63**(2010), no. 2, 245–270. MR**2651911** - Kenneth R. Davidson and Dilian Yang,
*Periodicity in rank 2 graph algebras*, Canad. J. Math.**61**(2009), no. 6, 1239–1261. MR**2588421**, DOI 10.4153/CJM-2009-058-0 - Kenneth R. Davidson and Dilian Yang,
*Representations of higher rank graph algebras*, New York J. Math.**15**(2009), 169–198. MR**2511133** - Cynthia Farthing, Paul S. Muhly, and Trent Yeend,
*Higher-rank graph $C^*$-algebras: an inverse semigroup and groupoid approach*, Semigroup Forum**71**(2005), no. 2, 159–187. MR**2184052**, DOI 10.1007/s00233-005-0512-2 - Adam Hanley Fuller,
*Finitely correlated representations of product systems of $C^*$-correspondences over $\Bbb N^k$*, J. Funct. Anal.**260**(2011), no. 2, 574–611. MR**2737414**, DOI 10.1016/j.jfa.2010.10.004 - Matthew Kennedy,
*Wandering vectors and the reflexivity of free semigroup algebras*, J. Reine Angew. Math.**653**(2011), 47–73. MR**2794625**, DOI 10.1515/CRELLE.2011.019 - Matthew Kennedy,
*The structure of an isometric tuple*, Proc. Lond. Math. Soc. (3)**106**(2013), no. 5, 1157–1177. MR**3066752**, DOI 10.1112/plms/pds065 - David W. Kribs and Stephen C. Power,
*Free semigroupoid algebras*, J. Ramanujan Math. Soc.**19**(2004), no. 2, 117–159. MR**2076898** - David W. Kribs and Stephen C. Power,
*The analytic algebras of higher rank graphs*, Math. Proc. R. Ir. Acad.**106A**(2006), no. 2, 199–218. MR**2266827**, DOI 10.3318/PRIA.2006.106.2.199 - Alex Kumjian and David Pask,
*Higher rank graph $C^\ast$-algebras*, New York J. Math.**6**(2000), 1–20. MR**1745529** - Alex Kumjian, David Pask, and Aidan Sims,
*Homology for higher-rank graphs and twisted $C^\ast$-algebras*, J. Funct. Anal.**263**(2012), no. 6, 1539–1574. MR**2948223**, DOI 10.1016/j.jfa.2012.05.023 - Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy,
*Harmonic analysis of operators on Hilbert space*, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR**2760647**, DOI 10.1007/978-1-4419-6094-8 - David Pask, Iain Raeburn, Mikael Rørdam, and Aidan Sims,
*Rank-two graphs whose $C^*$-algebras are direct limits of circle algebras*, J. Funct. Anal.**239**(2006), no. 1, 137–178. MR**2258220**, DOI 10.1016/j.jfa.2006.04.003 - Gelu Popescu,
*Isometric dilations for infinite sequences of noncommuting operators*, Trans. Amer. Math. Soc.**316**(1989), no. 2, 523–536. MR**972704**, DOI 10.1090/S0002-9947-1989-0972704-3 - Gelu Popescu,
*von Neumann inequality for $(B({\scr H})^n)_1$*, Math. Scand.**68**(1991), no. 2, 292–304. MR**1129595**, DOI 10.7146/math.scand.a-12363 - Gelu Popescu,
*Non-commutative disc algebras and their representations*, Proc. Amer. Math. Soc.**124**(1996), no. 7, 2137–2148. MR**1343719**, DOI 10.1090/S0002-9939-96-03514-9 - Stephen C. Power,
*Classifying higher rank analytic Toeplitz algebras*, New York J. Math.**13**(2007), 271–298. MR**2336241** - Iain Raeburn, Aidan Sims, and Trent Yeend,
*Higher-rank graphs and their $C^*$-algebras*, Proc. Edinb. Math. Soc. (2)**46**(2003), no. 1, 99–115. MR**1961175**, DOI 10.1017/S0013091501000645 - Charles J. Read,
*A large weak operator closure for the algebra generated by two isometries*, J. Operator Theory**54**(2005), no. 2, 305–316. MR**2186356** - Dilian Yang,
*Endomorphisms and modular theory of 2-graph $C^*$-algebras*, Indiana Univ. Math. J.**59**(2010), no. 2, 495–520. MR**2648076**, DOI 10.1512/iumj.2010.59.3973 - Dilian Yang,
*Type III von Neumann algebras associated with 2-graphs*, Bull. Lond. Math. Soc.**44**(2012), no. 4, 675–686. MR**2967236**, DOI 10.1112/blms/bdr132

## Additional Information

**Adam H. Fuller**- Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130
- MR Author ID: 916003
- ORCID: 0000-0002-9002-0501
- Email: afuller7@math.unl.edu
**Dilian Yang**- Affiliation: Department of Mathematics $\&$ Statistics, University of Windsor, Windsor, Ontario N9B 3P4, Canada
- MR Author ID: 668010
- Email: dyang@uwindsor.ca
- Received by editor(s): June 5, 2013
- Published electronically: March 26, 2015
- Additional Notes: The second author was partially supported by an NSERC Discovery grant.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 6199-6224 - MSC (2010): Primary 47L55, 47L30, 47L75, 46L05
- DOI: https://doi.org/10.1090/S0002-9947-2015-06385-5
- MathSciNet review: 3356934