*Proof.*
(a)

(b)Let , then is a closed ideal in . By , . For all , so , hence and thus . Therefore, .

(c) is a unitary is equal to is a isomorphism in the sense is a surjection which preserves inner product. Denote by , then ;

thus is a unitary.

Â â—»

*Proof.* For all positive linear functional
on
,
is a positive linear functional on
and thus bounded,
i.e. for some positive constant
.

Every bounded linear functional on is sum of 4 positive linear functionals on by Jordan Decomposition, thus for all for some positive constant . Hence for some positive constant by the Principle of Uniform Boundedness, i.e. is bounded.Â â—»

*Proof.* (a) If
is invertible, then
, thus
is strictly positive.

Conversely, if is strictly positive, then . Since the set of all invertible elements is open, . Hence there exists some such that is invertible, thus is invertible.

(b)If is strictly positive in , that is , then there exists a sequence in such that Hence Therefore, and .

Conversely, if , then for all then there exists a sequence in such that , hence Therefore, every rank-one projection belongs to . Since all the rank-one projections is the total subset of , .

(c)Suppose is a positive functional such that , then so Hence for all , Therefore, for .Â â—»

*Proof.* Suppose
is a basis for
,
such taht
for all
. and a bounded net
in
converges strongly to an operator
. For every
, there exists a finite subset
of
such that
and there exists
such that
Therefore,
for some constant
. Hence
.Â â—»

*Proof.* (1) Suppose
and
, then
and
, so
.Hence