nLab Hilbert bimodule



Functional analysis

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algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



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The notion of Hilbert C *C^\ast-bimodule adapts the notion of bimodules over associative algebras to operator algebra/C-star-algebra theory.



For A,BA,B \in C*Alg two C-star algebras, an (A,B)(A,B)-Hilbert C *C^\ast-bimodule (or just Hilbert bimodule, for short) is

  • a right BB-Hilbert C*-module (N,,)(N, \langle -,-\rangle);

  • equipped with a further left AA-representation A(N)A \to \mathcal{B}(N) by adjointable operators, hence such that a,=,a *\langle a -,- \rangle = \langle -,a^\ast -\rangle for all aAa \in A.

A isomorphism between two (A,B)(A,B)-bimodules (N 1,,)(N 2,, 2)(N_1, \langle -,-\rangle) \to (N_2, \langle -,-\rangle_2) is a linear operator N 1N 2N_1\to N_2 which is unitary with respect to , 2\langle -,-\rangle_2.


Given an (A,B)(A,B)-Hilbert bimodule (N 1,, 1)(N_1, \langle -,-\rangle_1) and a (B,C)(B,C)-Hilbert bimodule (N 2,, 2)(N_2, \langle -,-\rangle_2), the tensor product of Hilbert bimodules N 1 BN 2N_1 \otimes_B N_2 is the (A,C)(A,C)-Hilbert bimodule obtained from the ordinary (algebraic) tensor product of modules over \mathbb{C} N 1 N 2N_1 \otimes_{\mathbb{C}} N_2 by

  1. equipping it with the CC-valued inner product defined by

    ξ 1η 1,ξ 2η 2η 1,ξ 1,ξ 2η 2. \left\langle \xi_1 \otimes \eta_1 , \xi_2 \otimes \eta_2\right\rangle \coloneqq \left\langle \eta_1, \left\langle \xi_1, \xi_2\right\rangle \cdot \eta_2 \right\rangle \,.
  2. forming the quotient by the submodule of elements vv for which v,v=0\langle v,v\rangle = 0;

  3. forming the completion of this quotient with respect to the induced norm.


Def. really does yield a kind of tensor product over BB: elements of the form

vbwvbw v \cdot b \otimes w - v \otimes b \cdot w

are in the submodule that it being divided out, because

vbwvbw,vbwvbw w,vb,vbw +bw,v,vbw w,vb,vbw bw,v,vbw =(1+111)w,vb,vbw =0, \begin{aligned} \langle v \cdot b \otimes w - v \otimes b \cdot w \;,\; v \cdot b \otimes w - v \otimes b \cdot w \rangle & \coloneqq \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle \\ & + \langle b \cdot w, \langle v,v\rangle \cdot b \cdot w\rangle \\ & - \langle w, \langle v\cdot b, v \rangle \cdot b \cdot w\rangle \\ & - \langle b \cdot w, \langle v, v \cdot b \rangle \cdot w\rangle \\ & = (1+1-1-1) \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle \\ & = 0 \end{aligned} \,,

where we use that by definition the left actions are required to have adjoints, so that for instance

bw,v,vbw =w,b *v,vbt =w,vb,vbw \begin{aligned} \langle b \cdot w , \langle v,v\rangle \cdot b \cdot w\rangle & = \langle w , b^\ast \cdot \langle v,v\rangle \cdot b \cdot t\rangle \\ & = \langle w , \langle v \cdot b,v \cdot b\rangle \cdot w\rangle \end{aligned}

There is a (2,1)-category C *Alg bC^\ast Alg_b whose

(Buss-Zhu-Meyer 09)


An (,B)(\mathbb{C},B)-Hilbert C *C^\ast-bimodule is equialently just an BB-Hilbert C*-module.

An (A,)(A, \mathbb{C})-Hilbert C *C^\ast-bimodule is equivalently just as representation of a C-star-algebra.




For instance

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.