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Hilbert C^*-Module


The notion of a Hilbert C^*-module is a generalization of the notion of a Hilbert space. The first use of such objects was made by Kaplansky (1953). The research on Hilbert C^*-modules began in the 1970s in the work of the induced representations ofC^*-algebras by Rieffel (1974) and the doctoral dissertation of Paschke (1973). Hilbert C^*-modules are useful tools in AW^*-algebra theory, theory of operator algebras, operator K-theory, group representation theory, and theory of operator spaces. It is also used to study Morita equivalence of C^*-algebras, K-theory of C^*-algebras, C^*-algebra quantum group (Lance 1995, Wegge-Olsen 1993).

A pre-Hilbert module over a C^*-algebra A is a complex linear space E which is a left A-module (and lambda(ax)=(lambdaa)x=a(lambdax) where lambda in C, a in A, and x in E) equipped with an A-valued inner product <·,·>:E×E->A satisfying:

1. <x,x>>=0,

2. <x,x>=0 iff x=0,

3. <x+lambday,z>=<x,z>+lambda<y,z>,

4. <y,x>=<x,y>^*,

5. <ax,y>=a<x,y>.

A pre-Hilbert A-module is called a Hilbert A-module or Hilbert C^*-module over A, if it is complete with respect to the norm ||x||=||<x,x>||^(1/2). If the closed linear span of the set {<x,y>:x,y in E} is dense in A then E is called full. For example every C^*-algebra A is a full Hilbert A-module whenever we define <x,y>=xy^*.


See also

Hilbert Space

This entry contributed by Mohammad Sal Moslehian

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References

Kaplansky, I. "Modules Over Operator Algebras." Amer. J. Math. 75, 839-858, 1953.Lance, E. C. Hilbert C-*-Modules: A Toolkit for Operator Algebraists. Cambridge, England: Cambridge University Press, 1995.Paschke, W. L. "Inner Product Modules Over B*-Algebras." Trans. Amer. Math. Soc. 182, 443-468, 1973.Rieffel, M. A. "Morita Equivalence Representations of C^*-Algebras." Adv. in Math. 13, 176-257, 1974.Wegge-Olsen, N. E. K-Theory and C-*-Algebras: A Friendly Approach. Oxford, England: Oxford University Press, 1993.

Cite this as:

Moslehian, Mohammad Sal. "Hilbert C^*-Module." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HilbertC-Star-Module.html

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