QUASINILPOTENT EQUIVALENCE IN BANACH ALGEBRAS Heinrich Raubenheimer - PDF document

QUASINILPOTENT EQUIVALENCE IN BANACH ALGEBRAS Heinrich Raubenheimer University of Johannesburg 1

Let A be a Banach algebra and a, b ∈ A . Con- sider operators L a and R b defined on A as fol- lows: L a x = ax and R b x = xb for all x ∈ A . 2

Definition 1 Let A be a Banach algebra and a, b ∈ A . Define || ( L a − R b ) n 1 || 1 /n . ρ ( a, b ) = lim sup 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 ) . 3

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 . 4

|| ( L a − R b ) n 1 || 1 /n = 0 ρ ( a, b ) = lim sup ? ⇒ n and d ( a, b ) = max { ρ ( a, b ) , ρ ( b, a ) } = 0 ⇒ ? 5

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 ) . 6

? ⇒ d ( a, b ) = 0 . 7

• 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. 8

Proposition 2 Let A be a Banach algebra and a, b ∈ A . If a − b is a commuting quasinilpotent then d ( a, b ) = 0 . 9

Theorem 2 Let A be a Banach algebra and a, b ∈ A . Then ab = ba and d ( a, b ) = 0 if and only if a − b is a commuting quasinilpotent. 10

Theorem 3 Let A be a Banach algebra and a, b ∈ A . Then ab = ba and d ( a, b ) = 0 if and only 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 ( x 1 , x 2 ) = (0 , − x 1 ) and S ( x 1 , x 2 ) = ( x 2 , 0) for all ( x 1 , x 2 ) ∈ Y . Then d ( S, T ) = 0 , ST � = TS and S − T is not a commuting quasinilpotent. 11

Let A be a semisimple Banach algebra and a ∈ A . Define the rank of a by rank( a ) = sup #( σ ( ax ) \{ 0 } ) . x ∈ A An element A is said to be of maximal finite rank if rank( a ) = #( σ ( a ) \{ 0 } ) . 12

Theorem 4 Let A be a Banach algebra and a, b ∈ A . Then ab = ba and d ( a, b ) = 0 if and only 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 . 13

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 } ). 14

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 . 15

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 of σ ( a ) . If d ( a, b ) = 0 then a = b . 16