QUASINILPOTENT EQUIVALENCE IN BANACH ALGEBRAS Heinrich Raubenheimer - - PDF document

QUASINILPOTENT EQUIVALENCE IN BANACH ALGEBRAS Heinrich Raubenheimer University of Johannesburg

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Let A be a Banach algebra and a, b ∈ A. Con- sider operators La and Rb defined on A as fol- lows: Lax = ax and Rbx = xb for all x ∈ A.

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Definition 1 Let A be a Banach algebra and a, b ∈ A. Define ρ(a, b) = lim sup

n

||(La − Rb)n1||1/n.

• In general the numbers ρ(a, b) and ρ(b, a)

are different.

• If a and b commute then

ρ(a, b) = ρ(b, a) = r(a − b).

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Definition 2

• Let A be a Banach algebra

and a, b ∈ A. Define d(a, b) = max{ρ(a, b), ρ(b, a)}. The function d is called the spectral semidis- tance from a to b.

• The elements a and b are called quasinilpo-

tent equivalent if d(a, b) = 0.

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ρ(a, b) = lim sup

n

||(La − Rb)n1||1/n = 0 ⇒ ? and d(a, b) = max{ρ(a, b), ρ(b, a)} = 0 ⇒ ?

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Proposition 1 Let A be a Banach algebra and a, b ∈ A. If ρ(a, b) = 0 then r(a) = r(b). Theorem 1 Let A be a Banach algebra and a, b ∈ A. If d(a, b) = 0 then σ(a) = σ(b).

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? ⇒ d(a, b) = 0.

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• All quasinilpotent elements in A are quasinilpo-

tent equivalent.

• If a, b ∈ A and σ(a) = σ(b) = {λ} for some

λ ∈ C then d(a, b) = 0.

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Proposition 2 Let A be a Banach algebra and a, b ∈ A. If a − b is a commuting quasinilpotent then d(a, b) = 0.

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Theorem 2 Let A be a Banach algebra and a, b ∈ A. Then ab = ba and d(a, b) = 0 if and

• nly if a − b is a commuting quasinilpotent.

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Theorem 3 Let A be a Banach algebra and a, b ∈ A. Then ab = ba and d(a, b) = 0 if and

• nly if a − b is a commuting quasinilpotent.

Example 1 Let X be a Banach space and Y = X ⊕ X. Define operators T and S on Y by T(x1, x2) = (0, −x1) and S(x1, x2) = (x2, 0) for all (x1, x2) ∈ Y . Then d(S, T) = 0, ST = TS and S − T is not a commuting quasinilpotent.

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Let A be a semisimple Banach algebra and a ∈

• A. Define the rank of a by

rank(a) = sup

x∈A

#(σ(ax)\{0}). An element A is said to be of maximal finite rank if rank(a) = #(σ(a)\{0}).

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Theorem 4 Let A be a Banach algebra and a, b ∈ A. Then ab = ba and d(a, b) = 0 if and

• nly if a − b is a commuting quasinilpotent.

Theorem 5 Let A be a semisimple Banach al- gebra and 0 = a ∈ A a maximal finite rank el- ement and b ∈ A. Then d(a, b) = 0 if and only if a − b is a commuting quasinilpotent. Theorem 6 Let A be a semisimple Banach al- gebra with a, b ∈ A. If both a and b are of maximal finite rank and d(a, b) = 0 then a = b.

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Theorem 7 Let A be a semisimple Banach al- gebra and 0 = a ∈ A a maximal finite rank el- ement and b ∈ A. Then d(a, b) = 0 if and only if a − b is a commuting quasinilpotent. Theorem 8 (Brits) Let A be a semisimple Ba- nach algebra with a ∈ Soc A and suppose a assumes its rank on comm (a). If b ∈ A then d(a, b) = 0 if and only if a − b is a commuting quasinilpotent.

• a assumes its rank on comm(a) if there is

y ∈comm(a) such that rank(a) = #(σ(ay)\{0}).

• a is of maximal finite rank if rank(a) =

#(σ(a)\{0}) = #(σ(a · 1)\{0}).

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Theorem 9 Let A be a semisimple Banach al- gebra with a, b ∈ A. If both a and b are of maximal finite rank and d(a, b) = 0 then a = b. Theorem 10 (Brits) Let A be a semisimple Banach algebra with a, b ∈ Soc A. If a and b assume their respective ranks on comm( a) and comm( b) and if d(a, b) = 0 then a = b.

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Theorem 11 Let A be a C∗− algebra and let a be a normal element of A with finite spectrum. If b ∈ A, then d(a, b) = 0 if and only if a − b is a commuting quasinilpotent. Theorem 12 (Brits) Let A be a C∗−algebra and suppose both a and b are normal and sup- pose 0 is the only possible accumulation point

• f σ(a). If d(a, b) = 0 then a = b.

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