Nonself-adjoint Semicrossed Products by Abelian Semigroups

Fuller, Adam Hanley

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 768-782

Résumé

Let $S$ be the semigroup $S=\sum\nolimits_{i=1}^{\oplus k}{{{S}_{i}}}$ , where for each $i\in I,{{S}_{i}}$ is a countable subsemigroup of the additive semigroup ${{\mathbb{R}}_{+}}$ containing 0. We consider representations of $S$ as contractions ${{\left\{ {{T}_{s}} \right\}}_{s\in S}}$ on a Hilbert space with the Nica-covariance property: $T_{s}^{*}{{T}_{t}}={{T}_{t}}T_{s}^{*}$ whenever $t\wedge s=0$ . We show that all such representations have a unique minimal isometric Nica-covariant dilation.This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of $S$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms. We conclude by calculating the ${{C}^{*}}$ -envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

Détail

DOI : 10.4153/CJM-2012-051-8

Mots-clés : 47L55, 47A20, 47L65, semicrossed product, crossed product, C*-envelope, dilations

Bibliographie

[1] [1] Arveson, W. B., Operator algebras and measure preserving automorphisms. Acta Math. 118(1967), 95–109. Google Scholar | DOI

[2] [2] Arveson, W. B., Subalgebras of C*-algebras. Acta Math. 123(1969), 141–224. Google Scholar | DOI

[3] [3] Arveson, W. B., Subalgebras of C*-algebras. II. Acta Math. 128(1972), 271–308. Google Scholar | DOI

[4] [4] Davidson, K. R. and Katsoulis, E. G., Semicrossed products of simple C*-algebras. Math. Ann. 342(2008), 515–525. Google Scholar | DOI

[5] [5] Davidson, K. R., Dilating covariant representations of the non-commutative disc algebras. J. Funct. Anal. 259(2010), 817–831. Google Scholar | DOI

[6] [6] Davidson, K. R., Biholomorphisms of the unit ball of Cn and semicrossed products. In: Operator theory live, Theta Ser. Adv. Math. 12, Theta, Bucharest, 2010, 69–80. Google Scholar

[7] [7] Davidson, K. R., Semicrossed products of the disc algebra. Proc. Amer. Math. Soc., to appear. http://dx.doi.org/10.1090/S0002-9939-2012-11348-6 Google Scholar

[8] [8] Davidson, K. R., Dilation theory, commutant lifting and semicrossed products. Doc. Math. 16(2011), 781–868. Google Scholar

[9] [9] Dritschel, M. and McCullough, S., Boundary representations of families of representations of operator algebras and spaces. J. Operator Theory 53(2005), 159–197. Google Scholar

[10] [10] Duncan, B. L., C*-envelopes of universal free products and semicrossed products for multivariable dynamics. Indiana Univ. Math. J. 57(2008), 1781–1788. http://dx.doi.org/10.1512/iumj.2008.57.3273 Google Scholar

[11] [11] Duncan, B. L. and Peters, J. R., Operator algebras and representations from commuting semigroup actions. Preprint, arxiv:1008.2244v1. Google Scholar

[12] [12] Fowler, N. J., Discrete product systems of Hilbert bimodules. Pacific J. Math. 204(2002), 335–375. Google Scholar | DOI

[13] [13] Hamana, M., Injective envelopes of operator systems. Publ. Res. Inst. Math. Sci. 15(1979), 773–785. Google Scholar | DOI

[14] [14] Kakariadis, E. T. A., Semicrossed products of C*-algebras and their C*-envelopes. Preprint, arxiv:1102.2252v2. Google Scholar

[15] [15] Kakariadis, E. T. A. and Katsoulis, E. G., Semicrossed products of operator algebras and their C*-envelopes. J. Funct. Anal. 262(2012), 3108–3124. Google Scholar | DOI

[16] [16] Laca, M., From endomorphisms to automorphisms and back: dilations and full corners. J. London Math. Soc. (2) 61(2000), 893–904. Google Scholar | DOI

[17] [17] Laca, M. and Raeburn, I., Semigroup crossed products and the Toeplitz algebras of nonabelian groups. J. Funct. Anal. 139(1996), 415–440. Google Scholar | DOI

[18] [18] Ling, K. and Muhly, P. S., An automorphic form of Ando's theorem. Integral Equations Operator Theory 12(1989), 424–434. Google Scholar | DOI

[19] [19] McAsey, M., Muhly, P. S. and Saito, K., Nonselfadjoint crossed products (invariant subspaces and maximality). Trans. Amer. Math. Soc. 248(1979), 38–409. Google Scholar | DOI

[20] [20] Mlak, W., Unitary dilations in case of ordered groups. Ann. Polon. Math. 17(1966), 321–328. Google Scholar

[21] [21] Muhly, P. and Solel, B., An algebraic characterization of boundary representations. In: Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics, Birkhäuser Verlag, Basel, 1998, 189–196. Google Scholar

[22] [22] Murphy, G. J., Crossed products of C*-algebras by endomorphisms. Integral Equations Operator Theory 24(1996), 298–319. http://dx.doi.org/10.1007/BF01204603 Google Scholar

[23] [23] Nica, A., C*-algebras generated by isometries and Wiener–Hopf operators. J. Operator Theory 27(1992), 17–52. Google Scholar

[24] [24] Paulsen, V. I., Completely bounded maps and operator algebras. Cambridge Stud. Adv. Math. 78, Cambridge University Press, Cambridge, 2002. Google Scholar

[25] [25] Pedersen, G. K., C*-algebras and their automorphism groups. London Math. Soc. Monogr. 14, Academic Press, Inc., London–New York, 1979. Google Scholar

[26] [26] Peters, J., Semicrossed products of C*-algebras. J. Funct. Anal. 59(1984), 498–534. Google Scholar | DOI

[27] [27] Shalit, O. M., Dilation theorems for contractive semigroups. arxiv:1004.0723v1. Google Scholar

[28] [28] Shalit, O. M., Representing a product system representation as a contractive semigroup and applications to regular isometric dilations. Canad. Math. Bull. 53(2010), 550–563. Google Scholar | DOI

[29] [29] Solel, B., Representations of product systems over semigroups and dilations of commuting CP maps. J. Funct. Anal. 235(2006), 593–618. Google Scholar | DOI

[30] [30] Solel, B., Regular dilations of representations of product systems. Math. Proc. R. Ir. Acad. 108(2008), 89–110. Google Scholar | DOI

[31] [31] Sz.-Nagy, B., Foias, C., Bercovici, H., and Kérchy, L., Harmonic analysis of operators on Hilbert space. Universitext. Springer, New York, 2010. Google Scholar

[32] [32] Timotin, D., Regular dilations and models for multicontractions. Indiana Univ. Math. J. 47(1998), 671–684. Google Scholar

[33] [33] Varopoulos, N. Th., On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. J. Funct. Anal. 16(1974), 83–100. Google Scholar | DOI