Math 7330: Functional Analysis
Fall 2002
Notes/Homework 6: Banach *Algebras
A. Sengupta
An
involution
*
on a complex algebra
B
is a map
*
:
B
→
B
for which
(i)
*
(
a
+
b
) =
*
a
+
*
b
for all
a, b
∈
B
(ii)
*
(
λa
) =
λ
*
a
for all
λ
∈
C
and
a
∈
B
(iii) (
xy
) =
y
*
x
*
for all
x, y
∈
B
(iv) (
x
*
)
*
=
x
for all
x
∈
B
. An element
a
∈
B
is
hermitian
if
a
=
a
*
.
On a complex
Banach
algebra we also require an involution
*
to satisfy
(v)

xy
 ≤ 
x

y

for all
x, y
∈
B
.
Observe that for the identity
e
, we have
e
*
=
ee
*
and so taking
*
of this we get
(
e
*
)
*
= (
e
*
)
*
e
*
, which says
e
=
ee
*
. Thus
e
=
e
*
A
B*algebra
is a complex Banach algebra
B
on which there is an involution
*
for
which

xx
*

=

x

2
for all
x
∈
B
1. Let
B
be a complex Banach algebra with involution.
(i) Show that
if
B
is a B*algebra then

x

=

x
*

for all
x
∈
B
(ii) Suppose

y
*
y

=

y

2
for all
y
∈
B
. Show that

y

=

y
*

for all
y
∈
B
.
1
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(iii) Suppose

y
*
y

=

y

2
for all
y
∈
B
. Show that

xx
*

=

x

2
for all
x
∈
B
.
2. Let
B
be a B*algebra.
(i) Show that if
y
∈
B
is hermitian and
s
is any real number then

se
+
iy

2
=

s
2
e
+
y
2

(ii) Show that
e
+
iy
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 Spring '08
 Davidson,M
 Algebra, Exponential Function, Hilbert space, C*algebra

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