Algebra Valued Fields on Noncommutative Spaces Defined by C*-Algebras
Let ℋ_{S}(n) be a complex Hilbert space of dimension n, and *A*_{S}(n)
its C*-algebra of linear operators acting on it.
Let T be a normal operator in *A*_{S}(n); T is guaranteed to be diagonalizable
and possess an orthonormal eigenset that can serve as a basis
|t_{k}> for ℋ_{S}(n).
Let *A*_{F} an algebra over the complex field, i.e., *A*_{F} is a complex vector
space with a multiplication '*' that can be specified as follows:
* is a binary operation on *A*_{F} so that, if α_{k} is a basis of the
vector space,
α_{i} * α_{j} = Σ f_{ij}^{k} α_{k}
where f_{ij}^{k} are complex numbers, and structure constants of *A*_{F}.
To each eigenbasis element |t_{k}> for *A*_{S}(n) attach an element
α_{k} ∈ *A*_{F}.
Then to every element |ψ> ∈ ℋ_{S}(n) given by
Σ c^{k} |t_{k}>
there is attached, by linearity of both ℋ_{S}(n) and *A*_{F} an element
of *A*_{F} defined by the map
|ψ> → Σ c^{k} α_{k}
which defines an *A*_{F} valued function on ℋ_{S}(n), which then also defines
an *A*_{F} valued function on the "pure states" of of the C*-algebra *A*_{S}(n)
through the map of |ψ> to projection operators
M(ψ) := |ψ><ψ|
For any one of a collection of {|ψ>}
M(ψ_{a}) → Σ Σ c_{a}^{*k} c_{a}^{j} α^{†}_{k} α_{a}
where α^{†}_{k} is an element of the dual space of *A*_{F} as a vector space.
Again by linearity, since any state Ψ on *A*_{S}(n) is a convex linear combination
of pure states,
D(Ψ) = Σ p^{a} M(ψ_{a})
where
Σ p^{a} = 1, p^{a} ≥ 0
defines a density matrix, or state on *A*_{S}(n).
The map
D(Ψ) → Σ Σ Σ p^{a} c_{a}^{k*} c_{a}^{j} α_{k}^{†} α_{j}
defines an *A*_{F} valued function on the full forward cone of states of
*A*_{S}(n), C( *A*_{S}(n) ).
Any element of A ∈ *A*_{S}(n) can be expressed as a linear combination of
C( *A*_{S}(n) ), and so finally we have defined an *A*_{F} valued function on
all of *A*_{S}(n).
In terms of bundles, let *A*_{S}(n) be a base space and *A*_{F} the structure of
a fibre over *A*_{S}(n).
*A*_{S}(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 *A*_{F} as a vector space
through the constructed functional map of *A*_{S}(n) → *A*_{F}.
Though the representation of GL(n, C) on *A*_{S}(n) is fundamental and
irreducible, the associated representative action on *A*_{F} is not
necessarily irreducible.
Upon what does the irreducibility of the representative action on *A*_{F}
depend?

Footnotes
1.
2.
3.

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Created: 1997

Last Updated: May 28, 2000

Last Updated: June 30, 2004