*-Bimodules

Schmüdgen, Konrad

doi:10.1090/proc/15528

$*$-Bimodules
HTML articles powered by AMS MathViewer

by Konrad Schmüdgen PDF

Proc. Amer. Math. Soc. 149 (2021), 3923-3938 Request permission

Abstract:

A $*$-bimodule for a unital $*$-algebra $A$ is an $A$-bimodule $X$ which is a vector space with involution $x\mapsto x^+$ satisfying $(a\cdot x\cdot b)^+=b^+\cdot x^+\cdot b^+$ for $x\in X$ and $a,b\in A$. An algebraic model for $*$-bimodules is given. Hilbert space representations of $*$-bimodules are defined and studied. A GNS-like representation theorem is obtained.

References

Jean-Pierre Antoine, Atsushi Inoue, and Camillo Trapani, Partial $^*$-algebras and their operator realizations, Mathematics and its Applications, vol. 553, Kluwer Academic Publishers, Dordrecht, 2002. MR 1947892, DOI 10.1007/978-94-017-0065-8
T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
Gerd Lassner, Algebras of unbounded operators and quantum dynamics, Phys. A 124 (1984), no. 1-3, 471–479. Mathematical physics, VII (Boulder, Colo., 1983). MR 759198, DOI 10.1016/0378-4371(84)90263-2
Konrad Schmüdgen, Unbounded operator algebras and representation theory, Operator Theory: Advances and Applications, vol. 37, Birkhäuser Verlag, Basel, 1990. MR 1056697, DOI 10.1007/978-3-0348-7469-4
Konrad Schmüdgen, Noncommutative real algebraic geometry—some basic concepts and first ideas, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 325–350. MR 2500470, DOI 10.1007/978-0-387-09686-5_{9}
Konrad Schmüdgen, An invitation to unbounded representations of $*$-algebras on Hilbert space, Graduate Texts in Mathematics, vol. 285, Springer, Cham, [2020] ©2020. MR 4180683, DOI 10.1007/978-3-030-46366-3
Maria Fragoulopoulou and Camillo Trapani, Locally convex quasi $^*$-algebras and their representations, Lecture Notes in Mathematics, vol. 2257, Springer, Cham, [2020] ©2020. MR 4162276, DOI 10.1007/978-3-030-37705-2