minimal unitizations of algebras with additional structure
Given a (non-unital) algebra (http://planetmath.org/Algebra) there is a procedure to add an unit to it (parent entry (http://planetmath.org/Unitization)). When the algebra has some additional structure (topological structure, for example), it is often useful to endow the same structure on the minimal unitization of the algebra.
All the algebras are to be considered non-unital.
0.1 Topological Algebras
Let be a topological algebra algebra over a (topological) field . Let be its minimal unitization.
Then is a topological algebra with the product topology.
0.2 Normed and Banach Algebras
Let be a normed algebra over ( or ) with norm . Let be its minimal unitization.
Then is a normed algebra under the norm :
Moreover, if is a Banach algebra, then is a Banach algebra with the norm .
Let be a *-algebra over an involutory field (http://planetmath.org/InvolutaryRing) . Let be its minimal unitization.
Then is a *-algebra with involution given by:
0.4 Topological *-algebras, Normed *-algebras and Banach *-algebras
Let be a topological *-algebra over . Let be its minimal unitization.
Then is a topological *-algebra with the product topology and the involution defined above.
Also, if is a normed *-algebra (Banach -*algebra), then is also a normed *-algebra (Banach *-algebra) under the above involution and the norm .
Let be a -algebra (http://planetmath.org/CAlgebra) with norm . Let be its minimal unitization.
Then is -algebra under the norm :
This norm comes from regarding elements of as left on . The norm is to the norm .
|Title||minimal unitizations of algebras with additional structure|
|Date of creation||2013-03-22 17:46:29|
|Last modified on||2013-03-22 17:46:29|
|Last modified by||asteroid (17536)|
|Defines||minimal unitization of a topological algebra|
|Defines||minimal unitization of a Banach algebra|
|Defines||minimal unitization of a Banach-* algebra|
|Defines||minimal unitization of a -algebra|