THE SPECTRAL SEMIDISTANCE IN BANACH ALGEBRAS R BRITS Let A be a - PowerPoint PPT Presentation

THE SPECTRAL SEMIDISTANCE IN BANACH ALGEBRAS R BRITS

Let A be a Banach complex algebra with identity 1 . For elements a , b ∈ A the spectral semidistance between a and b is defined as follows: Denote the commutator C a , b := L a − R b , and then consider powers evaluated at 1 : n � n � C n � ( − 1) k a n − k b k . a , b 1 = k k =0 Writing � 1 / n � C n � � ρ ( a , b ) := lim sup a , b 1 n we define the spectral semidistance d ( a , b ) := sup { ρ ( a , b ) , ρ ( b , a ) } . The spectral semidistance is a semimetric and could be viewed as the noncommutative generalization of the distance induced by the spectral radius in the commutative case. If d ( a , b ) = 0 then a and b are called quasinilpotent equivalent.

The spectral semidistance between decomposable operators S , T ∈ L ( X ) can be formulated in terms of spectra via Vasilescu’s geometric formula: d ( S , T ) = sup { ∆( σ T ( x ) , σ S ( x )) : 0 � = x ∈ X } where σ S ( x ) and σ T ( x ) are, respectively, the local spectra of S and T at x ∈ X .

THEOREM Suppose σ ( a ) and σ ( b ) are finite with σ ( a ) = { λ 1 , . . . , λ n } , σ ( b ) = { β 1 , . . . , β k } . If { p 1 , . . . , p n } and { q 1 , . . . , q k } are the corresponding Riesz projections then ρ ( a , b ) = sup {| λ i − β j | : p i q j � = 0 } . (1) THEOREM Suppose σ ′ ( a ) and σ ′ ( b ) are discrete sets which cluster at 0 ∈ C , if anywhere. If σ ′ ( a ) = { λ 1 , λ 2 , . . . } and σ ′ ( b ) = { β 1 , β 2 , . . . } denote the nonzero spectral points of a and b, and if { p 1 , p 2 , . . . } and { q 1 , q 2 . . . } are the corresponding Riesz projections, then ̺ takes at least one of the following values: (i) ρ ( a , b ) = sup {| λ i − β j | : p i q j � = 0 } , or (ii) ρ ( a , b ) = | λ i | for some i ∈ N , or (iii) ρ ( a , b ) = | β i | for some i ∈ N . Moreover, ρ ( a , b ) = 0 if and only if the spectra and the corresponding Riesz projections of a and b coincide.

Let f be an entire function from C into a Banach algebra A . Then f has an everywhere convergent power series expansion ∞ � a n λ n , f ( λ ) = n =0 with coefficients a n belonging to A . Define a function M f ( r ) = sup � f ( λ ) � , r > 0 . | λ |≤ r DEFINITION The function f is said to be of finite order if there exists K > 0 and R > 0 such that M f ( r ) < e r K holds for all r > R. The infimum of the set of positive real numbers, K, such that the preceding inequality holds is called the order of f , denoted by ω f . If ω f = 1 then f is said to be of exponential order.

DEFINITION Suppose f is entire, and of finite order ω := ω f . Then f said to be of finite type if there exists L > 0 and R > 0 such that M f ( r ) < e Lr ω holds for all r > R. The infimum of the set of positive real numbers, L, such that the preceding inequality holds is called the type of f , denoted by τ f . It is well-known that the order and type of A -valued entire functions are given by the formulae � n log n � 1 � � � n n � a n � ω f ω f = lim sup and τ f = lim sup . log � a n � − 1 e ω f n n

Let a , b ∈ A , and define f : λ �→ e λ a e − λ b , λ ∈ C . The corresponding series expansion, valid for all λ ∈ C , is given by ∞ λ n C n a , b 1 f ( λ ) = e λ a e − λ b = � . n ! n =0 The important observations for us are the following: ◮ f is of order at most one ◮ If f is of order precisely one, then the type of f is given by ρ ( a , b ).

THEOREM Suppose d ( a , b ) = 0 , and suppose f is analytic on an open set U containing σ ( a ) = σ ( b ) . Then d ( f ( a ) , f ( b )) = 0 . THEOREM (P´ olya) Let f : C → A be an entire function from the field C into a Banach algebra A. If f is (norm) bounded over Z and if log M f ( r ) lim sup ≤ 0 , r r →∞ then f is constant. THEOREM ∈ σ ∗ ( a ) , then Let A be a Banach algebra, and let a , b ∈ A. If 0 / a = b if and only if a and b are quasinilpotent equivalent, and {� a n b − n � : n ∈ Z } is bounded. More generally, two elements, a and b, in a Banach algebra coincide if and only if they are are ∈ σ ∗ ( a ) such that quasinilpotent equivalent, and there exists α / {� ( α + a ) n ( α + b ) − n � : n ∈ Z } is bounded.

COROLLARY Let A be a C ∗ -algebra, and let a , b ∈ A be both self-adjoint or both be unitary . Then a = b if and only if a and b are quasinilpotent equivalent. THEOREM (Brits & Raubenheimer) Let A be a C*-algebra. If a and b are normal elements of A, and if 0 is the only possible accumulation point of σ ( a ) , then d ( a , b ) = 0 if and only if a = b. The above results can be improved to hold for arbitrary normal elements using the fact that the spectral semidistance is related to the growth characteristics of an entire function from C to A . For this we shall need a typical version of the Phragm´ en-Lindel¨ of device.

THEOREM (Phragm´ en-Lindel¨ of) Let u be a subharmonic function on the half-plane H := { λ ∈ C : Re λ > 0 } , such that for some constants A , B < ∞ u ( λ ) ≤ A + B | λ | , λ ∈ H . (2) If lim sup u ( λ ) ≤ 0 for all ζ ∈ ∂ H \{∞} (3) λ → ζ and if u ( t ) ( t ∈ R + ) lim sup = L , (4) t t →∞ then u ( λ ) ≤ L Re λ, λ ∈ H . (5) THEOREM Let A be a C ∗ -algebra and let a , b ∈ A be normal elements. Then a = b if and only if a and b are quasinilpotent equivalent.

SKETCH OF PROOF ◮ Define entire functions, f and g, from C into A by respectively f ( λ ) = e λ ia e − λ i ( b − b ∗ ) e − λ ia ∗ and g ( λ ) = e λ i ( a − a ∗ ) e − λ i ( b − b ∗ ) , and notice that r σ ( f ( λ )) = r σ ( g ( λ )) for all λ ∈ C . ◮ Then define C : λ �→ log r σ ( f ( λ )) which is subharmonic on C . ◮ Applying the growth characteristics it follows that, given ǫ > 0 arbitrary, there exists R ( ǫ ) > 0 such that for all r > R ( ǫ ) � � � e λ ia e − λ ib � � e λ ib ∗ e − λ ia ∗ � log r σ ( f ( λ )) ≤ log ≤ ǫ r whenever | λ | ≤ r, hence (2) holds.

◮ To establish (3): if ζ lies on the imaginary axis, then lim sup log r σ ( f ( λ )) = lim sup log r σ ( g ( λ )) ≤ 0 λ → ζ λ → ζ ◮ Finally, one also obtains log r σ ( f ( t )) lim sup = 0 , t t →∞ t > 0 whence it follows that r σ ( f ( λ )) = r σ ( g ( λ )) ≤ 1 for all λ ∈ { λ ∈ C : Re λ ≥ 0 } . ◮ Using a symmetric argument we can prove the same result for − H , and hence that r σ ( f ( λ )) = r σ ( g ( λ )) ≤ 1 for all λ ∈ C . ◮ Now define an entire function e λ i ( a − a ∗ ) e − λ i ( b − b ∗ ) − 1 � � � /λ if λ � = 0 h ( λ ) = i ( a − a ∗ ) − i ( b − b ∗ ) if λ = 0 .

◮ Since r σ ( g ( λ )) is bounded on C it follows that lim sup | λ |→∞ r σ ( h ( λ )) = 0. But r σ ( h ( λ )) is subharmonic on C and therefore, by Liouville’s Theorem (for subharmonic functions), it must be constantly zero on C ◮ In particular, we see that r σ ( i ( a − a ∗ ) − i ( b − b ∗ )) = 0. But i ( a − a ∗ ) − i ( b − b ∗ ) being self-adjoint it follows that a − a ∗ = b − b ∗ . ◮ Writing c := a − a ∗ = b − b ∗ we see that c commutes with both a and b from which we then obtain � a + a ∗ , b + b ∗ � a − c 2 , b − c � � d = d = d ( a , b ) = 0 . 2 2 2 So, using a preceding corollary, we get a + a ∗ = b + b ∗ and hence that a = b as required.