topological *-algebra

Definition (Involution) An involution on an algebra A over an involutory field ( F is a map *:AA:aa* such that for every {a,b}A and λF we have

  1. 1.


  2. 2.

    (ab)*=b*a* and

  3. 3.

    (λa+b)*=λ*a*+b*, where λ* denotes the involution ( of λ in F.

Definition (*-Algebra) A *-algebra is an algebra with an involution.

Definition (Topological *-algebra) A topological *-algebra is a *-algebra which is also a topological vector spaceMathworldPlanetmath such that its algebra multiplication and involution are continuous.

0.0.1 Remarks:

  • *-algebras are a particular of involutory rings.

  • The involutory field F is often taken as , where the involution is given by complex conjugation. In this case, condition 3 could be rewritten as:


  • Banach *-algebras are topological *-algebras.

Title topological *-algebra
Canonical name Topologicalalgebra
Date of creation 2013-03-22 14:45:38
Last modified on 2013-03-22 14:45:38
Owner HkBst (6197)
Last modified by HkBst (6197)
Numerical id 12
Author HkBst (6197)
Entry type Definition
Classification msc 22A30
Classification msc 16W80
Classification msc 16W10
Classification msc 46K05
Classification msc 46H35
Synonym topological *-algebra
Related topic BanachAlgebra
Related topic WeakHopfCAlgebra2
Related topic VonNeumannAlgebra
Defines involution *-algebra
Defines *-algebra