A note on cocycle-conjugate endomorphisms of von Neumann algebras

Floricel, Remus

doi:10.1090/S0002-9939-04-07314-9

A note on cocycle-conjugate endomorphisms of von Neumann algebras
HTML articles powered by AMS MathViewer

by Remus Floricel PDF

Proc. Amer. Math. Soc. 132 (2004), 2013-2018 Request permission

Abstract:

We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm {I}_{\infty }.$

References

William Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (1989), no. 409, iv+66. MR 987590, DOI 10.1090/memo/0409
Ola Bratteli, Palle E. T. Jorgensen, and Geoffrey L. Price, Endomorphisms of ${\scr B}({\scr H})$, Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 59, Amer. Math. Soc., Providence, RI, 1996, pp. 93–138. MR 1392986, DOI 10.1090/pspum/059/1392986
Donald Bures and Hong Sheng Yin, Shifts on the hyperfinite factor of type $\textrm {II}_1$, J. Operator Theory 20 (1988), no. 1, 91–106. MR 972183
Donald Bures and Hong Sheng Yin, Outer conjugacy of shifts on the hyperfinite $\textrm {II}_1$-factor, Pacific J. Math. 142 (1990), no. 2, 245–257. MR 1042044, DOI 10.2140/pjm.1990.142.245
Marie Choda, Square roots of the canonical shifts, J. Operator Theory 31 (1994), no. 1, 145–163. MR 1316989
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330, DOI 10.1007/BF01625776
M. Laca, Endomorphisms of $\scr B(\scr H)$ and Cuntz algebras, J. Operator Theory 30 (1993), no. 1, 85–108. MR 1302609
Robert T. Powers, An index theory for semigroups of $^*$-endomorphisms of ${\scr B}({\scr H})$ and type $\textrm {II}_1$ factors, Canad. J. Math. 40 (1988), no. 1, 86–114. MR 928215, DOI 10.4153/CJM-1988-004-3
Geoffrey L. Price, Shifts on the hyperfinite $\rm II_1$ factor, J. Funct. Anal. 156 (1998), no. 1, 121–169. MR 1632956, DOI 10.1006/jfan.1997.3225
Şerban Strătilă, Modular theory in operator algebras, Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells, 1981. Translated from the Romanian by the author. MR 696172