- #1

- 39

- 1

- Homework Statement
- C*-algebra which may or may not be unital.

- Relevant Equations
- involution and multiplication given by:

Let $$\mathcal{A}$$ be a $C*$-algebra which may or may not have a unit with norm $$||.||$$, and put $$\mathcal{\overline{A}} = \mathcal{A} \oplus \mathbb{C}$$ as a vector space with mupltiplication:

$$(a, \lambda) (b, \mu) = (ab + \lambda b + \mu a , \lambda \mu)$$,

$$(a, \lambda)^{*} = (a^{*}, \overline{\lambda}).$$

Then I must show that $$\mathcal{\overline{A}}$$ is a $$*$$-algebra with unit 1 = (0,1).

1)

(0,1) is the unit since:

$$(a, \lambda)(0,1) = (a0 + \lambda 0 + 1a , \lambda 1) = (a, \lambda)$$

2)

$$(A^{*})^{*} = A \forall \mathcal{\overline{A}}$$ since:

$$((a, \lambda)^{*})^{*} = (a^{*}, \overline{\lambda})^{*} = (a^{**}, \overline{\overline{\lambda}}) = (a, \lambda)$$.

However I am struggling a bit with showing that

3) $$(aA + bB)^{*} = (\overline{a} A^{*} + \overline{b} B^{*}) \ \forall A,B \in \overline{\mathcal{A}}, a,b \in \mathbb{C}$$

and

4) $$(AB)^{*} = B^{*} A^{*}$$

$$(a, \lambda) (b, \mu) = (ab + \lambda b + \mu a , \lambda \mu)$$,

$$(a, \lambda)^{*} = (a^{*}, \overline{\lambda}).$$

Then I must show that $$\mathcal{\overline{A}}$$ is a $$*$$-algebra with unit 1 = (0,1).

1)

(0,1) is the unit since:

$$(a, \lambda)(0,1) = (a0 + \lambda 0 + 1a , \lambda 1) = (a, \lambda)$$

2)

$$(A^{*})^{*} = A \forall \mathcal{\overline{A}}$$ since:

$$((a, \lambda)^{*})^{*} = (a^{*}, \overline{\lambda})^{*} = (a^{**}, \overline{\overline{\lambda}}) = (a, \lambda)$$.

However I am struggling a bit with showing that

3) $$(aA + bB)^{*} = (\overline{a} A^{*} + \overline{b} B^{*}) \ \forall A,B \in \overline{\mathcal{A}}, a,b \in \mathbb{C}$$

and

4) $$(AB)^{*} = B^{*} A^{*}$$