On logarithms of measures

Let A be a Banach algebra and let x ∈ A have the property that its spectrum does not separate 0 from inﬁnity. It is well known that x has a logarithm, i


Introduction
Throughout this paper A will be a complex Banach algebra with identity 1 and invertible group A −1 .We denote the spectrum of x ∈ A by σ (x) = {λ ∈ C : λ − x / ∈ A −1 } and the spectral radius of x by r (x) with r (x) = sup{| λ | : λ ∈ σ (x) }.The spectral radius of a Banach algebra element x can be calculated by the well known formula r (x) = lim n→∞ x n 1/n .If it is important to emphasize the relevant algebra A, then the spectrum and spectral radius of x ∈ A will be denoted by σ (x, A) and r (x, A) respectively.If an element x ∈ A has the property that σ (x) = {0}, or equivalently that r (x) = 0, then it is called quasinilpotent.We call a subspace I of A an ideal if I A ⊂ I and AI ⊂ I .By exp A we designate the set exp A = {e x : x ∈ A}.We will say x ∈ A has a logarithm if x ∈ exp A. For convenience we will denote the set {λ ∈ C : |λ| = 1 } by T. Suppose A is a Banach algebra and x ∈ A is invertible.If 0 belongs to the unbounded component of C\σ (x) (the complement of the spectrum), then we will say that the spectrum of x does not separate 0 from infinity.The following result in spectral theory of Banach algebras is well known: Theorem 1.1 [3,Theorem 3.3.6]Let A be a Banach algebra.Suppose x ∈ A has a spectrum which does not separate 0 from infinity.Then there exists y ∈ A such that x = e y .
We mention some corollaries of Theorem 1.1 that can assist one to identify logarithms in a Banach algebra.

Corollary 1.2 Let A be a Banach algebra and x
If there exists n ∈ N with x n = 1, then x has a logarithm.3.If the spectrum of x has at most 0 as a limit point, then λ − x has a logarithm for all λ / ∈ σ (x).
If X is a Banach space then the collection L(X ) of bounded operators on X is a Banach space with the operator norm.If one defines multiplication in L(X ) as composition of operators it becomes a Banach algebra with identity the identity operator.Let E be a complex Banach lattice.An operator T : E → E is called regular if it can be written as a linear combination over C of positive operators.The space of regular operators on E will be denoted by L r (E) and it is a subspace of L(E).When L r (E) is provided with the r -norm T r = inf{ S : S ∈ L(E), S ≥ 0, |T z| ≤ S|z| for all z ∈ E} it becomes a Banach algebra which contains the unit of L(E), see for instance the remarks preceding Lemma 1.1 in [1].The spectrum of a regular operator T in L r (E) is termed the o-spectrum of T and it is denoted by σ o (T ).This concept was introduced by Schaefer [9] and it was extensively investigated in [1,2].An operator T ∈ L r (E) is called r -compact if it can be approximated in the r -norm by operators of finite rank, [1].These operators will be denoted by K r (E).The center Z(E) of E consists of all operators T ∈ L(E) for which there exists c > 0 such that |T x| ≤ c|x| for all x ∈ E.
Let X be a locally compact group with identity e.By a measure on X we understand a complex regular Borel measure.By C 0 (X ) we denote the space of all continuous complex valued functions vanishing at infinity.It is well known that the space M(X ) of all bounded measures on X may be identified with the dual space of C 0 (X ), [5, C.18].Note that M(X ) is a vector space over the complex numbers C. Define for μ ∈ M(X ) the variation of μ by If we let μ = |μ|(X ), then μ is a norm on M(X ) and M(X ) is a Banach space with this norm.For μ, ν ∈ M(X ) the ordering μ ≤ ν, i.e., μ(E) ≤ ν(E) for all Borel sets E ⊂ X together with the variation norm induce the structure of a Banach lattice on M(X ).The positive cone in M(X ) will be denoted by M(X ) + .By the above identification a measure μ ∈ M(X ) corresponds to the linear functional This allows one to define a convolution product μ * ν for μ, ν ∈ M(X ) by If x ∈ X and if E ⊂ X is a Borel set, let δ x be the measure defined by With the variation norm and convolution product, M(X ) becomes a Banach algebra with identity δ e .If X is a abelian group, we will denote the identity by 0 and in this case it is well known that M(X ) is a commutative Banach algebra.

A counter example
In this section we provide a counter example to show that in general the converse of Theorem 1.1 is not true.Let X be a locally compact abelian group and let 1 ≤ p ≤ ∞.Consider the complex Banach space L p (X ), where the measure is Haar measure.The translation operator on L p (X ) associated with x ∈ X is the operator T x, p on L p (X ) defined by where f ∈ L p (X ) and t ∈ X a.e..If 1 < p < ∞ and if x ∈ X has infinite order, then by [6,Theorem 20.18] the spectrum of the operator T x, p in the Banach algebra L(L p (X )) is the set T. Hence the spectrum of T x, p in L(L p (X )) separates 0 from infinity.However, in view of [6,Theorem 20.19] there exists an operator A x, p ∈ L(L p (X )) with T x, p = e i A x, p , i.e., T x, p has a logarithm.

The spectrum of ı x
Let X be a locally compact abelian group and x ∈ X .We show how to calculate the spectrum of the measure δ x .The Banach algebra M(X ) is often represented in the Banach algebra L(L p (X )) via the mapping μ → T μ, p where T μ, The bounded operators T μ, p are called convolution operators.This representation has the inconvenience that the spectrum of μ in the Banach algebra M(X ) does not in general coincide with the spectrum of the operator T μ, p in the Banach algebra L(L p (X )).
By noting that T μ, p is positive whenever μ is positive, we know that every convolution operator is regular since every measure in M(X ) is a linear combination of positive measures.Thus, the mapping μ → T μ, p is actually a representation of M(X ) into the Banach algebra L r (L p (X )).W. Arendt proved in [1,Proposition 3.3] that if X is amenable, then the mapping μ → T μ, p is an isometric algebra and lattice isomorphism onto a full subalgebra of L r (L p (X )) for 1 ≤ p < ∞.This implies that the spectrum of μ in M(X ) coincides with the spectrum of T μ, p in L r (L p (X )), i.e. σ (μ) = σ 0 (T μ, p ). Theorem 3.1 Let X be a locally compact abelian group and let x ∈ X have infinite order and suppose μ = δ x .
Proof It is easy to see that T μ, p = T x, p for 1 ≤ p ≤ ∞.

Compact groups
Let X be a compact group.In this section we give an illustration of Corollary 1.2.To accomplish this we invoke convolution operators on L p (X ) with 1 ≤ p ≤ ∞.The spaces L p (X ) are understood relative to the left Haar measure m on X .By invoking the Radon Nikodym theorem, L 1 (X ) can be identified with an ideal in M(X ).In fact, L 1 (X ) is a closed Banach algebra ideal in M(X ).Hence, M(X )/L 1 (X ) is a Banach algebra under the quotient norm.The natural homomorphism q : M(X This means that a measure μ in M(X ) belongs to M 0 (X ) if the image of μ under the natural homomorphism in the quotient algebra M(X )/L 1 (X ) is quasinilpotent.Recall that an ideal I in a Banach algebra is called inessential if for every x ∈ I , the spectrum σ (x) of x has at most 0 as an accumulation point.
Proof We claim that the ideal ).Since the rnorm is finer than operator norm, T f •m, p is a compact operator.In view of Theorem 4.1, σ ( f • m) has at most 0 as a limit point.
In our next remarks we illustrate that there are many measures in M(X ) belonging to the set M 0 (X ). • is quasinilpotent and ν ∈ L 1 (X ), then μ + ν ∈ M 0 (X ).

Infinitely divisible probability measures
Let X be a locally compact group.We show how one can use our previous results to identify infinitely divisible probability measures: A measure μ in M(X ) is called a probability measure if μ ≥ 0 and μ(X ) = 1.A probability measure on X is said to be infinitely divisible if for each n ∈ N there exists a probability measure μ n with