Advances in Operator Theory

Operator algebras associated to modules over an integral domain

Benton Duncan

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We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.

Article information

Adv. Oper. Theory, Volume 3, Number 2 (2018), 374-387.

Received: 15 June 2017
Accepted: 20 October 2017
First available in Project Euclid: 15 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47L74
Secondary: 47L40: Limit algebras, subalgebras of $C^*$-algebras

semicrossed product integral domain module


Duncan, Benton. Operator algebras associated to modules over an integral domain. Adv. Oper. Theory 3 (2018), no. 2, 374--387. doi:10.15352/AOT.1706-1181.

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  • W. Arveson, Operator algebras and measure preserving automorphisms, Acta Math. 118 (1967), 95–109.
  • W. Arveson and K. Josephson, Operator algebras and measure preserving automorphisms. II, J. Funct. Anal. 4 (1969), 100–134.
  • N. Brown and N. Ozawa, $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008.
  • H. Choda, A correspondence between subgroups and subalgebras in a discrete $C^*$-crossed product, Math. Japonica 24 (1979), 225–229.
  • J. Cuntz and X. Li, The regular $C^*$-algebra of an integral domain, Quanta of maths, 149–170 Clay Math. Proc. 11 Amer. Math. Soc., Providence RI, 2010.
  • K. Davidson and E. Katsoulis, Isomorphisms between topological conjugacy algebras, J. Reine Angew. Math. 621 (2008), 29–51.
  • K. Davidson and E. Katsoulis, Semicrossed products of simple $C^*$-algebras, Math. Ann. 342 (2008), 515–525.
  • K. Davidson and E. Katsoulis, Operator algebras for multivariable dynamics, Mem. Amer. Math. Soc. 209 (2011), no. 982.
  • K. Davidson and E. Katsoulis, Dilation theory, commutant lifting, and semicrossed products, Doc. Math. 16 (2011), 781–868
  • K. Davidson, A. Fuller, and E. Kakariadis, Semicrossed products of operator algebras by semigroups, Memoirs Amer. Math. Soc. 239 (201X).
  • B. Duncan, Operator algebras associated to integral domains, New York J. Math. 19 (2013), 39–50.
  • B. Duncan and J. Peters, Operator algebras and representations from commuting semigroup actions, J. Operator Theory 74 (2015), 23–43.
  • A. Fuller, Nonself-adjoint semicrossed products by abelian semigroups, Canad. J. Math. 65 (2013), 768–782.
  • E. Kakariadis and E. Katsoulis, Isomorphism invariants for multivariable $C^*$-dynamics, J. Noncommut. Geom. 8 (2014), 771–787.
  • M. Landstad, D. Olesen, and G. Pedersen, Towards a Galois theory for crossed products of $C^*$-algebras, Math. Scand. 43 (1978), 311–321.
  • J. Peters, Semicrossed products of $C^*$-algebras, J. Funct. Anal. 59 (1984), 498–534.
  • J. Peters, The $C^*$-envelope of a semicrossed product and nest representations, Operator structures and dynamical systems, 197–215, Contemp. Math., 503, Amer. Math. Soc., Providence RI, 2009.
  • S. Roman, Advanced linear algebra, third edition, Graduate Texts in Mathematics, 135, Springer, New York, NY, 2008.