nLab
multiplier 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

Given a nonunital C*-algebra AA, the multiplier algebra M(A)M(A) is the maximal unital extension of AA in which AA is an essential ideal (an ideal having nonzero intersection with any other proper ideal). In the commutative case, every unital commutative extension in which AA is an essential ideal corresponds to a compactification of the spectrum Spec(A)Spec(A); M(A)M(A) is in that case the C *C^*-algebra of functions on the Stone–Čech compactification? β(Spec(A))\beta(Spec(A)).

Definition

The multiplier algebra M(A)M(A) of a (not necessarily unital) C *C^*-algebra AA is the C *C^*-algebra satisfying the following universal property: for any C *C^*-algebra BB containing AA as an ideal, there exists a unique **-homomorphism ϕ:BM(A)\phi:B\to M(A) such that ϕ\phi extends the identity homomorphism on AA and ϕ(A )={0}\phi(A^\perp)=\{0\}.

The multiplier algebra M(A)M(A) can be realized as the set of 2-sided multipliers in the enveloping von Neumann algebra of AA. If ABA\subset B, bBb\in B is a multiplier (operator) for AA if aA\forall a\in A, baAb a\in A and abAa b\in A.

Of course, if AA is already unital, then M(A)=AM(A) = A.

There are also larger local multiplier algebra?s M loc(A)M_{loc}(A).

Literature

  • eom: Gert K. Pedersen, multiplier algebra

  • wikipedia: multiplier algebra

  • Ch.A. Akemann, G.K. Pedersen, J. Tomiyama, Multipliers of C *C^*-algebras, J. Funct. Anal. 13 (1973) 277–301 MR470685

  • Paul Skoufranis, An introduction to multiplier algebras, (pdf)

  • Corran Webster, On unbounded operators affiliated with C *C^*-algebras, J. Operator Theory 51 (2004) 237-244 dvi pdf ps

We show that the multipliers of Pedersen’s ideal of a C *C^*-algebra AA correspond to the densely defined operators on AA which are affiliated with AA in the sense defined by Woronowicz, and whose domains contain Pedersen’s ideal. We also extend the theory of q-continuity developed by Akemann to unbounded operators and show that these operators correspond to self-adjoint operators affiliated with AA.

There is also a notion of multiplier Hopf algebra and even multiplier bimonoid and Hopf monoid:

Last revised on April 30, 2018 at 05:43:31. See the history of this page for a list of all contributions to it.