Taub–NUT space

Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space), in that they endow a linear space with an "inner product" which 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"). In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Aristide Rieffel, the latter in a paper which used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. 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, and groupoid C*-algebras.

Definitions

Inner-product A-modules

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

$\langle \cdot ,\cdot \rangle :E\times E\rightarrow A$

which satisfies the following properties:

• For all x, y, z in E, and α, β in C:

$\langle x,\alpha y+\beta z\rangle =\alpha \langle x,y\rangle +\beta \langle x,z\rangle$

(i.e. the inner product is linear in its second argument).

• For all x, y in E, and a in A:

$\langle x,ya\rangle =\langle x,y\rangle a$

• For all x, y in E:

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

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

• For all x in E:

$\langle x,x\rangle \geq 0$

and

$\langle x,x\rangle =0\iff x=0.$

(An element of a C*-algebra A is said to be positive if it is self-adjoint with non-negative spectrum.)

Hilbert A-modules

An analogue to the Cauchy-Schwarz inequality holds for an inner-product A-module E:

$\langle x,y\rangle \langle y,x\rangle \leq \Vert \langle x,x\rangle \Vert \langle y,y\rangle$

for x, y in E.

On the pre-Hilbert module E, define a norm by

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

The norm-completion of E, still denoted by E, is said to be a Hilbert A-module or a Hilbert C*-module over the C*-algebra 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 A on E is continuous: for all x in E

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

Similarly, if {e_λ} is an approximate unit for A (a net of self-adjoint elements of A for which ae_λ and e_λa tend to a for each a in A), then for x in E

$xe_{\lambda }\rightarrow x$

whence it follows that EA is dense in E, and x1 = x when A is unital.

Let

$\langle E,E\rangle =\operatorname {span} \{\langle x,y\rangle |x,y\in E\},$

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

Examples

Hilbert spaces

A complex Hilbert space H is a Hilbert C-module under its inner product, the complex numbers being a C*-algebra with an involution given by complex conjugation.

Vector bundles

If X is a locally compact Hausdorff space and E a vector bundle over X with a Riemannian metric g, then the space of continuous sections of E is a Hilbert C(X)-module. The inner product is given by

$\langle f,h\rangle (x):=g(f(x),h(x)).$

The converse holds as well: Every countably generated Hilbert C*-module over a commutative C*-algebra A = C(X) is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over X.

C*-algebras

Any C*-algebra A is a Hilbert A-module under the inner product <a,b> = a*b. By the C*-identity, the Hilbert module norm coincides with C*-norm on A.

The (algebraic) direct sum of n copies of A

$A^{n}=\oplus _{1}^{n}A$

can be made into a Hilbert A-module by defining

$\langle (a_{i}),(b_{i})\rangle =\sum a_{i}^{*}b_{i}.$

One may also consider the following elements in the countable direct product of A

${\mathcal {H}}_{A}=\{(a_{i})|\sum a_{i}^{*}a_{i}{\text{ converges in }}A\}.$

Given an inner product analogous to that on Aⁿ, the resulting Hilbert A-module is called the standard Hilbert module.

See also

• Operator algebra

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• 
  
  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
• Hilbert C*-Modules Home Page, a literature list