algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
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Hilbert module is an abbreaviation both for a Hilbert $C^\ast$-module (this entry) and the analogous notion of a Hilbert Q-module (see there), where $Q$ is a quantale (or a locale, in particular).
The notion of Hilbert $C^\ast$-module (or simply Hilbert module) is a generalization of the notion of Hilbert space where the algebra of complex numbers is replaced by a possibly more general C*-algebra $A$. In particular a Hilbert $A$-module has an inner product which takes values not in $\mathbb{C}$, but in $A$, and such that complex conjugation is replaced by the star-operation in $A$.
Hilbert $C^\ast$-modules naturally appear as modules over groupoid convolution algebras. Refined to Hilbert C*-bimodules they serve as generalized homomorphism between C*-algebras in noncommutative topology, and, when further equipped with a left weak Fredholm module as cocycles in KK-theory.
For $B \in$ C*Alg, a Hilbert C*-module over $B$ is
a complex vector space $H$;
equipped with an action of $B$ from the right;
equipped with a sesquilinear map (linear in the second argument)
(the $B$-valued inner product)
such that
$\langle -,-\rangle$ behaves like a positive definite inner product over $B$ in that for all $x,y \in H$ and $b \in B$ we have
$\langle x,y\rangle^\ast = \langle y,x\rangle$
$\langle x,x\rangle \geq 0$ (in the sense of positive elements in $B$)
$\langle x,x\rangle = 0$ precisely if $x = 0$;
$\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b$
$H$ is complete with respect to the norm
${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}$.
In addition to the explicit $B$-linearity in the second argument under right multiplicatojn
the axioms imply conjugate $B$-linearity in the first argument and under left multiplication
Because:
First of all we have:
An ordinary complex Hilbert space is a Hilbert $\mathbb{C}$-module.
The archetypical class of examples of Hilbert $C^\ast$-modules for commutative C*-algebras is the following. The general definition 1 may be understood as the generalization of the structure of this example to non-commutative C*-algebras. See also remark 3 below.
Let $X$ be a locally compact topological space and write $C_0(X)$ for its C*-algebra of continuous functions vanishing at infinity.
Let $E \to X$ be a fiber bundle of Hilbert spaces over $X$, hence a canonically associated bundle to a unitary group-principal bundle. Then the space $\Gamma_0(E)$ of continuous compactly supported sections is a Hilbert $C^\ast$-module over $C_0(X)$ with $C_0(X)$-valued inner product $\langle -,-\rangle$ the pointwise inner product in the Hilbert space fiber of $E$:
Every Hilbert $C_0(X)$-module arises, up to isomorphism, as in example 2.
Every $C^\ast$-algebra $A$ is a Hilbert $A$-module over itself when equipped by with the $A$-valued inner product given simply by
In view of the archetypical example 2, example 3 may be interpreted as exhibiting the trivial complex line bundle over whatever space $A$ is the $C^\ast$-algebra of functions on (an actual topological space if $A$ is a commutative C*-algebra or else the noncommutative topology defined as the formal dual of $A$).
For $A \in$ C*Alg, let $\ell^2 A$ be the space of those sequences $\{a_n \in A\}_{n \in \mathbb{N}}$ of elements in $A$ such that the series $\sum_n a_n^\ast a_n$ converges. This is a Hilbert $A$-module when equipped with the degreewise $A$-C*-representation, with the $A$-valued inner product
and after completion with under the induced norm.
This $\ell^2 A$ is sometimes called the standard Hilbert $A$-module over $A$.
In view of example 2 we may think of example 4 as exhibiting the trivial countably-infinite dimensional Hilbert space bundle over the space dual to $A$.
This is because the unitary group $U(\mathcal{H})$ of an infinite-dimensional separable Hilbert space $\mathcal{H}$ is contractible (by Kuiper's theorem), hence so is the classifying space, and so unitary $\mathcal{H}$-fiber bundles (over actual topological spaces) all trivializable. Since moreover $\mathcal{H} \simeq \ell^2(\mathbb{C})$ the Hilbert module of example 2 for the trivial $\mathcal{H}$-bundle over $C_0(X)$ is equivalent to $\ell^2(C_0(X))$. Example 4 generalizes this to arbitrary $C*$-algebras $A$.
For $A \in$ C*Alg and $H$ a Hilbert $A$-module, def. 1, a $\mathbb{C}$-linear operator $F \colon H \to H$ is called adjointable if there is an adjoint operator $F^\ast \colon H \to H$ with respect to the $A$-valued inner product in the sense that
The adjointable operators on a Hilbert $A$-module, def. 2, form a Banach star-algebra.
For $A$ itself regarded as a Hilbert $A$-module as in example 3, this is the multiplier algebra of $A$.
For $H_1, H_2$ two Hilbert $C^\ast$-modules, an adjointable operator $T \colon H_1 \to H_2$, def. 2, is of finite rank if it is of the form
for vectors $v_i \in H_1$ and $w_i \in H_2$. $T$ is called a generalized compact operator if it is in the norm-closure of finite-rank operators.
Typically one writes $\mathcal{K}(H_1, H_2)$ for the space of generalized complact operators.
An operator $F \colon H_1 \to H_2$ is called a generalized Fredholm operator if there exists an operator $S \colon H_2 \to H_1$ (then called a parametrix for $F$) such that both
$F \circ S - id_{H_2}$ and $S \circ F - id_{H_1}$
are compact operators according to def. 3.
Hilbert $C^\ast$-modules were introduced by Irving Kaplansky? in
Contemporary references are