algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
The notion of Hilbert -bimodule adapts the notion of bimodules over associative algebras to operator algebra/C-star-algebra theory.
For C*Alg two C-star algebras, an -Hilbert -bimodule (or just Hilbert bimodule, for short) is
a right -Hilbert C*-module ;
equipped with a further left -representation by adjointable operators, hence such that for all .
A isomorphism between two -bimodules is a linear operator which is unitary with respect to .
Given an -Hilbert bimodule and a -Hilbert bimodule , the tensor product of Hilbert bimodules is the -Hilbert bimodule obtained from the ordinary (algebraic) tensor product of modules over by
equipping it with the -valued inner product defined by
forming the quotient by the submodule of elements for which ;
forming the completion of this quotient with respect to the induced norm.
Def. really does yield a kind of tensor product over : elements of the form
are in the submodule that it being divided out, because
where we use that by definition the left actions are required to have adjoints, so that for instance
There is a (2,1)-category whose
objects are C*-algebras;
1-morphisms are Hilbert bimodules;
2-morphisms are isomorphisms of Hilbert bimodules;
vertical composition is the evident composition of intertwiners;
horizontal composition is the tensor product of Hilbert bimodules, def. .
An -Hilbert -bimodule is equialently just an -Hilbert C*-module.
An -Hilbert -bimodule is equivalently just as representation of a C-star-algebra.
(…)
For instance
Sergio Doplicher, Claudia Pinzari, Rita Zuccante, The -algebra of a Hilbert Bimodule [arXiv:funct-an/9707006]
Nik Weaver, Hilbert bimodules with involution, Canad. Math. Bull. 44 3 (2001) 355–369 [arXiv:math/9908119, pdf]
The tensor product of Hilbert bimodules and the induced 2-category structure is discussed in
Last revised on January 13, 2024 at 08:10:45. See the history of this page for a list of all contributions to it.