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unitization of a C-star-algebra

Contents

Context

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Noncommutative geometry

Contents

Idea

The generalization of the process of one-point compactification of possibly non-compact topological spaces from topology to non-commutative topology is the process of adding units to possibly non-unital C*-algebras, thought of as formal duals of non-commutative spaces in noncommutative topology.

Definition

Definition

For (A, A) (A, {\Vert \cdot \Vert_{A}}) a non-unital C*-algebra, its unitisation is the C * C^{\ast} -algebra whose underlying vector space is the direct sum

A +A A^{+} \coloneqq A \oplus \mathbb{C}

of A A with the field of complex numbers, equipped with the multiplication law

(a 1z 1)(a 2z 2)(a 1a 2+z 2a 1+z 1a 2)z 1z 2, (a_{1} \oplus z_{1}) \cdot (a_{2} \oplus z_{2}) \coloneqq (a_{1} a_{2} + z_{2} a_{1} + z_{1} a_{2}) \oplus z_{1} z_{2},

and the involution

(az) *a *z¯ (a \oplus z)^\ast \coloneqq a^{\ast} \oplus \overline{z}

(complex conjugation is taking place on the right). This really is a C *C^\ast-algebra. The norm can be characterised as

az A +=L a+zId A (A)sup bA,b A1ab+zb A. {\Vert a \oplus z \Vert}_{A^{+}} = {\Vert L_{a} + z \cdot \operatorname{Id}_{A} \Vert}_{\mathcal{B}(A)} \coloneqq \sup_{b \in A, \Vert b \Vert_{A} \leq 1} \Vert a b + z b \Vert_{A}.

if AA does not have a multiplicative unit. (Note that the operatornorm vanishes on 1 A1 1_A\oplus -1_{\mathbb{C}} if AA does have a multiplicative unit). For C *C^\ast-algebras with a multiplicative unit one can show that the norm equals

az A +=max{a+z1 A,|z|}\Vert a\oplus z \Vert_{A^+} = \max\{\|a+z1_A\|,|z|\}

Examples

Example

For XX a locally compact Hausdorff topological space and C(X)C(X) its possibly non-unital C*-algebra of functions, then

(C 0(X)) +C 0(X)C(X +) (C_0(X))^+ \simeq C_0(X) \oplus \mathbb{C} \simeq C(X^+)

this is equivalently the unital C *C^\ast-algebra of functions on the one-point compactification of XX.

There is hence a canonical projection

i *:C(X +) i^\ast \colon C(X^+) \to \mathbb{C}

The topological K-theory of XX is the kernel of the induced map in operator K-theory

K(X)ker(i *)K(C 0(X)). K(X) \simeq ker(i^\ast) \simeq K(C_0(X)) \,.

Properties

The point at infinity

Remark

The unitisation of AA comes with a canonical projection homomorphism of C*-algebras

i A *:A + i^\ast_A \colon A^+ \to \mathbb{C}

given by

(a+z)z. (a + z) \mapsto z \,.

Dually this corresponds to the inclusion of the “point at infinity”.

K-theory with compact support on non-unital C *C^\ast-algebras

The map i A *:A +i^\ast_A \colon A^+ \to \mathbb{C} of remark induces a homomorphism of operator K-theory groups of the form

K(i A *):K(A +)K(). K(i^\ast_A) \colon K(A^+) \to K(\mathbb{C}) \simeq \mathbb{Z} \,.
Definition

The kernel of this map is the operator K-theory of the original possibly non-unital C *C^\ast-algebra AA:

K(A)ker(K(i A *)). K(A) \coloneqq ker(K(i^\ast_A)) \,.

Heuristically it is clear that this is the “compactly suppported” K-theory of the possibly non-compact non-commutative space given by the algebra AA. This statement has been made precise for instance in (Emerson 07, theorem 3.8)):

Proposition

For XX a locally compact Hausdorff topological space, there is a natural isomorphism

K(X)[X,Fred] cs, K(X) \simeq [X,Fred]_{cs} \,,

where on the right we homotopy classes of maps of compact support into the classifying space for K-theory (space of Fredholm operators).

References

Last revised on November 25, 2017 at 13:50:55. See the history of this page for a list of all contributions to it.