# Question 1

*Let be a Banach algebra such that for all the implication*

* or *

*holds. Let , be linear mappings from to itself such that for all ,*

*, , and .*

*Show that and are necessarily continuous.*

# Question 2

*Let be a unital C*-algebra.*

## (a)

*If are positive elements of , show that .*

### Solution (Wills)

For elements of a unital algebra :

If then so that

Now if , for any , . Hence and the result follows (note that need not be hermitian)

## (b)

*If is an invertible element of , show that for a unique unitary . Give an example of an element of for some Hilbert space that cannot be written as the product of a unitary times a positive operator.*

## (c)

*Show that if , then iff is unitary.*

### Solution

Suppose first that is unitary. Then . Also

that is is unitary also and hence

Suppose now that is *not *unitary…

# Question 3

*Let be a locally compact Hausdorff space, and suppose that the C*-algebra is generated by a sequence of projections . Show that the hermitian element *

*generates .*

## Solution

In the first instance, the only projection functions are the characteristic functions for . For now I am going to assume that with compact. Then each of the is as required (eh — actually they’re not continuous!).

A polynomial in the is significantly simplified because for . Also if is finite, then

Hence, if these projections generate then any may be approximated in the supremum norm by:

where the for finite .

Hence we must now simultaneously approximate to on using polynomials in (not quite – sometimes the overlap). In the first instance

…

# Question 4

*We shall see in the next chapter that all closed ideals in C*-algebras are necessarily self-adjoint. Give an example of an ideal in the C*-algebra that is not self-adjoint.*

# Question 5

*Let be an isometric linear map between unital *-algebras and such that (for all ) and . Show that .*

# Question 6

*Let be a unital C*-algebra.*

## (a)

*If ** and **, show that ** and **.*

*and*

*, show that**and**.*### Solution

Working under the slightly stronger hypothesis that * *(this guarantees convergence of ), we note that is closed (??). Now consider:

With the strengthening of the assumptions this series converges and closed, this sum of positive terms converges in . Hence . Taking roots on both sides we get the result.

## (b)

*For all , show that *

# Question 7

*Let be a unital C*-algebra.*

## (a)

*If , show that the map*

*, *

*is differentiable and that *

### Solution (Wills)

.

If , then .

## (b)

*Let be a closed vector subspace of which is unitarily invariant in the sense that for all unitaries . Show that if and .*

### Solution (Wills)

If a closed subspace is unitarily invariant. Let and make an additional assumption that (). Then

.

Hence, and also . Now if then (since is closed???). Therefore and hence .

## (c)

*Deduce that the closed linear span of the projections in has the property that and implies that .*

### Solution

First we prove that if is a projection and if is a unitary then is a projection.

, also

Now it’s just a simple calculation to show:

.

Hence satisfies the hypotheses of the last exercise and hence we are done.

# Question 8 (Fuglede’s Theorem)

*Let be a normal element of a C*-algebra , and an element commuting with . Show that also commutes with .*

## Solution (Hinted by Murphy)

Let be the unitisation of and let be defined by

From Exercise 2.7 this map is holmorphic with derivative at 0

As commutes with , commutes with any continuous function of (or does this need to be polynomial – we don’t have compactness or a Hausdorff space either! Murphy suggests a different route certainly. However we do know that is normal). Now

Hence and hence constant. Therefore and we have as required

# Exercise 9

*If is an ideal of , show that it is self-adjoint.*

## Solution

From Theorem 2.4.7 we know that contains . Suppose has a basis . Define as the projection onto . Then is of finite rank and hence in the ideal.

Now let . Consider the operators . All of these are in the ideal and furthermore as and is closed, we have that . That is is self-adjoint

# Exercise 10

Let .

## (a)

Show that is a left topological divisor in iff it is not bounded below (cf. Exercise 1.11)

### Solution

From exercise 1.11 we know that:

- An element in unital Banach algebra is a
*left topological divisor*if there is a sequence of unit vectors such that . Equivalently where . - Left topological divisors are
*not*invertible.

Hence assume that is a left-topological divisor. Then is not invertible and hence not bounded below.

Assume that is not bounded below. Then for any sequence converging to zero, there exists a sequence such that:

By homogeneity of the norm the sequence may be chosen to be unit vectors.

Let be the sequence of linear projections onto . Then

.

Hence ; hence and so is a left-topological divisor.

## (b)

Define

.

This set is called the *approximate point spectrum of *because iff there is a sequence of unit vectors of such that . Show that is a closed subset of containing .

## (c)

Show that is bounded below iff it is left-invertible in .

## (d)

Show that if is normal.

# Question 11

Let be a normal operator with spectral resolution of the identity .

What the hell is a spectral resolution of the identity?

## (a)

Show that admits an invariant closed vector subspace other than and if .

## (b)

If is an isolated point of , show that and that is an eigenvalue of .

# Question 12

An operator on is *subnormal *if there is a Hilbert space containing as a closed vector subspace and there exists a normal operator on such that is invariant for , and is the restriction of . We call a *normal extension of .*

## (a)

Show that the unilateral shift is a non-normal subnormal operator.

### Solution

The unilateral shift is a non-normal subnormal operator vis the bilateral shift on a Hilbert Space with orthonormal basis where is the subspace with orthonormal basis .

## (b)

Show that if is subnormal, then .

## (c)

A normal extension of a subnormal operator is a *minimal *normal extension if the only closed vector subspace of reducing and containing is itself. Show that admits a minimal normal extension. In the case that is a minimal normal extension, show that is the closed linear span of all (, ).

## (d)

Show that if and are minimal normal extensions of , then there exists a unitary operator such that (so that there is obly one minimal normal extension).

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