# topological $*$-algebra

Definition (Involution) An involution on an algebra $A$ over an involutory field (http://planetmath.org/InvolutaryRing) $F$ is a map $\cdot^{*}:A\to A:a\mapsto a^{*}$ such that for every $\{a,b\}\subset A$ and $\lambda\in F$ we have

1. 1.

$a^{**}=a$,

2. 2.

$(ab)^{*}=b^{*}a^{*}$ and

3. 3.

$(\lambda a+b)^{*}=\lambda^{*}a^{*}+b^{*}$, where $\lambda^{*}$ denotes the involution (http://planetmath.org/InvolutaryRing) of $\lambda$ 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 space 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 $\mathbb{C}$, where the involution is given by complex conjugation. In this case, condition 3 could be rewritten as:

3.$\;(\lambda a+b)^{*}=\overline{\lambda}a^{*}+b^{*}$

• Banach *-algebras are topological $*$-algebras.

Title topological $*$-algebra Topologicalalgebra 2013-03-22 14:45:38 2013-03-22 14:45:38 HkBst (6197) HkBst (6197) 12 HkBst (6197) Definition msc 22A30 msc 16W80 msc 16W10 msc 46K05 msc 46H35 topological *-algebra BanachAlgebra WeakHopfCAlgebra2 VonNeumannAlgebra involution $*$-algebra *-algebra