Hilbert C*-module

Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces (which are themselves generalisations of Euclidean space), in that they endow a linear space with an "inner product" that takes values in a C*-algebra. Hilbert C*-modules were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital"). ^[1] In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke ^[2] and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. ^[3] Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, ^[4] and provide the right framework to extend the notion of Morita equivalence to C*-algebras. ^[5] They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, ^{[6] [7]} and groupoid C*-algebras.

Definitions

Inner-product C*-modules

Let ${\displaystyle A}$ be a C*-algebra (not assumed to be commutative or unital), its involution denoted by ${\displaystyle {}^{*}}$. An inner-product ${\displaystyle A}$-module (or pre-Hilbert ${\displaystyle A}$-module) is a complex linear space ${\displaystyle E}$ equipped with a compatible right ${\displaystyle A}$-module structure, together with a map

${\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{A}:E\times E\rightarrow A}$

that satisfies the following properties:

• For all ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ in ${\displaystyle E}$, and ${\displaystyle \alpha }$, ${\displaystyle \beta }$ in ${\displaystyle \mathbb {C} }$:

${\displaystyle \langle x,y\alpha +z\beta \rangle _{A}=\langle x,y\rangle _{A}\alpha +\langle x,z\rangle _{A}\beta }$

(i.e. the inner product is ${\displaystyle \mathbb {C} }$-linear in its second argument).

• For all ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$, and ${\displaystyle a}$ in ${\displaystyle A}$:

${\displaystyle \langle x,ya\rangle _{A}=\langle x,y\rangle _{A}a}$

• For all ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$:

${\displaystyle \langle x,y\rangle _{A}=\langle y,x\rangle _{A}^{*},}$

from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).

• For all ${\displaystyle x}$ in ${\displaystyle E}$:

${\displaystyle \langle x,x\rangle _{A}\geq 0}$

in the sense of being a positive element of A, and

${\displaystyle \langle x,x\rangle _{A}=0\iff x=0.}$

(An element of a C*-algebra ${\displaystyle A}$ is said to be positive if it is self-adjoint with non-negative spectrum.) ^{[8] [9]}

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product ${\displaystyle A}$-module ${\displaystyle E}$: ^[10]

${\displaystyle \langle x,y\rangle _{A}\langle y,x\rangle _{A}\leq \Vert \langle y,y\rangle _{A}\Vert \langle x,x\rangle _{A}}$

for ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$.

On the pre-Hilbert module ${\displaystyle E}$, define a norm by

${\displaystyle \Vert x\Vert =\Vert \langle x,x\rangle _{A}\Vert ^{\frac {1}{2}}.}$

The norm-completion of ${\displaystyle E}$, still denoted by ${\displaystyle E}$, is said to be a Hilbert ${\displaystyle A}$-module or a Hilbert C*-module over the C*-algebra ${\displaystyle A}$. The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

The action of ${\displaystyle A}$ on ${\displaystyle E}$ is continuous: for all ${\displaystyle x}$ in ${\displaystyle E}$

${\displaystyle a_{\lambda }\rightarrow a\Rightarrow xa_{\lambda }\rightarrow xa.}$

Similarly, if ${\displaystyle (e_{\lambda })}$ is an approximate unit for ${\displaystyle A}$ (a net of self-adjoint elements of ${\displaystyle A}$ for which ${\displaystyle ae_{\lambda }}$ and ${\displaystyle e_{\lambda }a}$ tend to ${\displaystyle a}$ for each ${\displaystyle a}$ in ${\displaystyle A}$), then for ${\displaystyle x}$ in ${\displaystyle E}$

${\displaystyle xe_{\lambda }\rightarrow x.}$

Whence it follows that ${\displaystyle EA}$ is dense in ${\displaystyle E}$, and ${\displaystyle x1_{A}=x}$ when ${\displaystyle A}$ is unital.

Let

${\displaystyle \langle E,E\rangle _{A}=\operatorname {span} \{\langle x,y\rangle _{A}\mid x,y\in E\},}$

then the closure of ${\displaystyle \langle E,E\rangle _{A}}$ is a two-sided ideal in ${\displaystyle A}$. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that ${\displaystyle E\langle E,E\rangle _{A}}$ is dense in ${\displaystyle E}$. In the case when ${\displaystyle \langle E,E\rangle _{A}}$ is dense in ${\displaystyle A}$, ${\displaystyle E}$ is said to be full. This does not generally hold.

Examples

Hilbert spaces

Since the complex numbers ${\displaystyle \mathbb {C} }$ are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space ${\displaystyle {\mathcal {H}}}$ is a Hilbert ${\displaystyle \mathbb {C} }$-module under scalar multipliation by complex numbers and its inner product.

Vector bundles

If ${\displaystyle X}$ is a locally compact Hausdorff space and ${\displaystyle E}$ a vector bundle over ${\displaystyle X}$ with projection ${\displaystyle \pi \colon E\to X}$ a Hermitian metric ${\displaystyle g}$, then the space of continuous sections of ${\displaystyle E}$ is a Hilbert ${\displaystyle C(X)}$-module. Given sections ${\displaystyle \sigma ,\rho }$ of ${\displaystyle E}$ and ${\displaystyle f\in C(X)}$ the right action is defined by

${\displaystyle \sigma f(x)=\sigma (x)f(\pi (x)),}$

and the inner product is given by

${\displaystyle \langle \sigma ,\rho \rangle _{C(X)}(x):=g(\sigma (x),\rho (x)).}$

The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra ${\displaystyle A=C(X)}$ is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over ${\displaystyle X}$. ^{[ citation needed ]}

C*-algebras

Any C*-algebra ${\displaystyle A}$ is a Hilbert ${\displaystyle A}$-module with the action given by right multiplication in ${\displaystyle A}$ and the inner product ${\displaystyle \langle a,b\rangle =a^{*}b}$. By the C*-identity, the Hilbert module norm coincides with C*-norm on ${\displaystyle A}$.

The (algebraic) direct sum of ${\displaystyle n}$ copies of ${\displaystyle A}$

${\displaystyle A^{n}=\bigoplus _{i=1}^{n}A}$

can be made into a Hilbert ${\displaystyle A}$-module by defining

${\displaystyle \langle (a_{i}),(b_{i})\rangle _{A}=\sum _{i=1}^{n}a_{i}^{*}b_{i}.}$

If ${\displaystyle p}$ is a projection in the C*-algebra ${\displaystyle M_{n}(A)}$, then ${\displaystyle pA^{n}}$ is also a Hilbert ${\displaystyle A}$-module with the same inner product as the direct sum.

The standard Hilbert module

One may also consider the following subspace of elements in the countable direct product of ${\displaystyle A}$

${\displaystyle \ell _{2}(A)={\mathcal {H}}_{A}={\Big \{}(a_{i})|\sum _{i=1}^{\infty }a_{i}^{*}a_{i}{\text{ converges in }}A{\Big \}}.}$

Endowed with the obvious inner product (analogous to that of ${\displaystyle A^{n}}$), the resulting Hilbert ${\displaystyle A}$-module is called the standard Hilbert module over ${\displaystyle A}$.

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert ${\displaystyle A}$-module ${\displaystyle E}$ there is an isometric isomorphism ${\displaystyle E\oplus \ell ^{2}(A)\cong \ell ^{2}(A).}$ ^[11]

See also

• Operator algebra

Notes

1. Kaplansky, I. (1953). "Modules over operator algebras". American Journal of Mathematics . 75 (4): 839–853. doi:10.2307/2372552. JSTOR   2372552.
2. Paschke, W. L. (1973). "Inner product modules over B*-algebras". Transactions of the American Mathematical Society . 182: 443–468. doi:10.2307/1996542. JSTOR   1996542.
3. Rieffel, M. A. (1974). "Induced representations of C*-algebras". Advances in Mathematics . 13 (2): 176–257. doi: 10.1016/0001-8708(74)90068-1 .
4. Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. Theta Foundation. 4: 133–150.
5. Rieffel, M. A. (1982). "Morita equivalence for operator algebras". Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 38: 176–257.
6. Baaj, S.; Skandalis, G. (1993). "Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres". Annales Scientifiques de l'École Normale Supérieure . 26 (4): 425–488. doi:10.24033/asens.1677.
7. Woronowicz, S. L. (1991). "Unbounded elements affiliated with C*-algebras and non-compact quantum groups". Communications in Mathematical Physics. 136 (2): 399–432. Bibcode:1991CMaPh.136..399W. doi:10.1007/BF02100032. S2CID   118184597.
8. Arveson, William (1976). An Invitation to C*-Algebras. Springer-Verlag. p. 35.
9. In the case when ${\displaystyle A}$ is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to ${\displaystyle A}$.
10. This result in fact holds for semi-inner-product ${\displaystyle A}$-modules, which may have non-zero elements ${\displaystyle A}$ such that ${\displaystyle \langle x,x\rangle _{A}=0}$, as the proof does not rely on the nondegeneracy property.
11. Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. ThetaFoundation. 4: 133–150.

References

• Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.

External links

• Weisstein, Eric W. "Hilbert C*-Module". MathWorld .
• Hilbert C*-Modules Home Page, a literature list