Algebra Valued Fields on Noncommutative Spaces Defined by C*-Algebras

Let ℋS(n) be a complex Hilbert space of dimension n, and AS(n) its C*-algebra of linear operators acting on it.

Let T be a normal operator in AS(n); T is guaranteed to be diagonalizable and possess an orthonormal eigenset that can serve as a basis |tk> for ℋS(n).

Let AF an algebra over the complex field, i.e., AF is a complex vector space with a multiplication '*' that can be specified as follows: * is a binary operation on AF so that, if αk is a basis of the vector space,

          αi * αj  =  Σ fijk αk

where fijk are complex numbers, and structure constants of AF.

To each eigenbasis element |tk> for AS(n) attach an element αk ∈ AF.

Then to every element |ψ> ∈ ℋS(n) given by

     Σ ck |tk>

there is attached, by linearity of both ℋS(n) and AF an element of AF defined by the map

     |ψ>  →  Σ ck αk

which defines an AF valued function on ℋS(n), which then also defines an AF valued function on the "pure states" of of the C*-algebra AS(n) through the map of |ψ> to projection operators

     M(ψ)  :=  |ψ><ψ|

For any one of a collection of {|ψ>}

     M(ψa)  →  Σ Σ ca*k caj α†k αa

where α†k is an element of the dual space of AF as a vector space.

Again by linearity, since any state Ψ on AS(n) is a convex linear combination of pure states,

     D(Ψ)  =  Σ pa M(ψa)

where

     Σ pa  =  1,  pa ≥ 0

defines a density matrix, or state on AS(n).

The map

     D(Ψ)  →  Σ Σ Σ  pa cak* caj αk† αj

defines an AF valued function on the full forward cone of states of AS(n), C( AS(n) ).

Any element of A ∈ AS(n) can be expressed as a linear combination of C( AS(n) ), and so finally we have defined an AF valued function on all of AS(n).

In terms of bundles, let AS(n) be a base space and AF the structure of a fibre over AS(n).

AS(n) always supports the group of basis substitutions isomorphic to GL(n, C). The adjoint action of this group, hence any of its subgroups, induces a representative action on AF as a vector space through the constructed functional map of AS(n) → AF.

Though the representation of GL(n, C) on AS(n) is fundamental and irreducible, the associated representative action on AF is not necessarily irreducible.

Upon what does the irreducibility of the representative action on AF depend?




Footnotes 



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Created: 1997
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Last Updated: June 30, 2004