Banach Algebras and Spectral Theory

2016 ◽  
pp. 16-49
Keyword(s):  
Spectral Theory ◽  
Banach Algebras

Spectral theory in ordered Banach algebras

Positivity ◽  
2016 ◽  
Vol 21 (2) ◽  
pp. 755-786 ◽  
Author(s):  
Sonja Mouton ◽  
Heinrich Raubenheimer
Keyword(s):  
Spectral Theory ◽  
Banach Algebras

scholarly journals Automatic continuity of positive functionals on topological involution algebras

1981 ◽  
Vol 23 (2) ◽  
pp. 265-281 ◽  
Author(s):  
P.G. Dixon
Keyword(s):  
Banach Algebra ◽  
Spectral Theory ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Automatic Continuity ◽  
Topological Algebras ◽  
Positive Functional ◽  
Open Questions ◽  
Positive Functionals ◽  
Locally Convex

This paper surveys the known results on automatic continuity of positive functionals on topological *-algebras and then shows how two theorems on Banach *-algebras extend to complete metrizable topological *-algebras. The two theorems concerned are Loy's theorem on separable Banach *-algebras A with centre Z such that AZ is of countable codimension and Varopoulos' result on Banach *-algebras with bounded approximate identity. Both theorems have the conclusion that all positive functionals on such algebras are continuous. The extension of the second theorem requires the algebra to be locally convex and the approximate identity to be ‘uniformly bounded’. Neither extension requires the algebra to be LMC. This means that the proof of the first theorem is quite different from the corresponding Banach algebra result (which used spectral theory). The proof of the second is closer to the previously known LMC version, but actually neater by being more general. It is also shown that the well-known estimate of |f(a*ba)| for a positive functional f on a Banach *-algebra may be obtained without the usual use of spectral theory. The paper concludes with a list of open questions.

Generalized Spectral Theory in Complex Banach Algebras

1985 ◽  
Vol 37 (6) ◽  
pp. 1211-1236
Author(s):  
G. N. Hile ◽  
W. E. Pfaffenberger
Keyword(s):  
Banach Algebra ◽  
Unit Circle ◽  
Compact Subset ◽  
Spectral Theory ◽  
Complex Plane ◽  
Banach Algebras ◽  
Complex Banach Algebra ◽  
Nonempty Compact ◽  
Nonempty Compact Subset

Let A be an element of a complex Banach algebra with identitI. The ordinary spectrum of A, sp(A), consists of those points z in the complex plane such that A — zI has no inverse in . If Q is any other element of , we define spQ(A), the spectrum of A relative to Q, or Q-spectrum of A, as those points z such that has no inverse in . Thus if Q = 0 the Q-spectrum of A is the same as the ordinary spectrum of A.The generalized notion of spectrum, spQ(A), retains many of the properties of the ordinary spectrum, particularly when A and Q commute and the ordinary spectrum of Q does not meet the unit circle. Under these conditions the Q-spectrum of A is a nonempty compact subset of the plane, and if both sp(A) and sp(Q) are finite (or countable), so is spQ(A).

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