Vidav-Palmer's theorem

The set of Vidav-Palmer , named after Ivan Vidav and Theodore W. Palmer , is a mathematical sentence from the branch of functional analysis . It characterizes the C * algebras among the Banach algebras and, as a corollary, enables a further characterization among all Banach - * algebras, which leads to a weakening of the usual C * condition. An essential aid is the generalization of the concept of the self-adjoint element to the concept of the Hermitian element (see below) with the help of the numerical value range .

Hermitian elements

Let it be a complex Banach algebra with one element . For will ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle a\in A}$

${\displaystyle V(A,a):=\{f(a);\,f\in A',f(e)=1=\|f\|\}\subset \mathbb {C} }$

referred to as the numerical range of values ​​of the element . One calls Hermitian if and is noted as the set of Hermitian elements. One can show that is a real Banach space and that the following statements are equivalent for a : ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle V(A,a)\subset \mathbb {R} }$ ${\displaystyle H(A)}$ ${\displaystyle H(A)}$ ${\displaystyle a\in A}$

• ${\displaystyle a\in H(A)}$, that is, is Hermitian. ${\displaystyle a}$
• ${\displaystyle \lim _{a\searrow 0}{\frac {1}{\alpha }}(\|1+\alpha a\|-1)\,=\,0}$
• ${\displaystyle \|\exp(i\alpha a)\|=1}$for all real numbers . ${\displaystyle \alpha }$

For the formation of , it should be noted that the associated exponential series converges in the Banach algebra . ${\displaystyle \exp(i\alpha a)}$ ${\displaystyle A}$

According to a theorem of AM Sinclair , the spectral radius of a Hermitian element agrees with its norm . This means that the convex hull of the spectrum corresponds to the numerical range of values. The latter is also known as Vidav's lemma and was previously proved by Vidav without the Sinclair theorem mentioned. Both proofs use functional theoretic tools, in particular the Phragmén-Lindelöf theorem .

Formulation of the sentence

Vidav-Palmer's theorem is:

• Be a complex Banach algebra with one element and let it apply . Then defined for an involution that makes a C * algebra. ${\displaystyle A}$ ${\displaystyle A=H(A)+i\cdot H(A)}$ ${\displaystyle (x+iy)^{*}\,:=\,x-iy}$ ${\displaystyle x,y\in H(A)}$ ${\displaystyle A}$

The theorem originally proved by Vidav contained the additional requirement that must apply to all ; Palmer has shown that this is dispensable. ${\displaystyle x^{2}\in H(A)}$ ${\displaystyle x\in H(A)}$

Inference

With Vidav-Palmer's theorem, the following characterization of the C * algebras can be proven, which originally goes back to James Glimm and Richard Kadison :

• A complex Banach algebra with an involution * is a C * -algebra if and only if holds for all . ${\displaystyle A}$ ${\displaystyle \|a^{*}a\|=\|a^{*}\|\|a\|}$ ${\displaystyle a\in A}$

Vidav-Palmer's theorem actually only delivers this result for Banach algebras with a unit, the version without a unit goes back to BJ Vowden . The condition in the above sentence is formally weaker than the usual C * condition for all . The theorem therefore shows that the weaker condition does not establish a new class of Banach algebras. ${\displaystyle \|a^{*}a\|=\|a\|^{2}}$ ${\displaystyle a\in A}$

Individual evidence

1. ^ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Cambridge University Press (1971), ISBN 0-521-07988-8 , Chapter 1, §5, Lemma 2
2. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862 , §10, Theorem 17
3. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862 , §38, Theorem 14
4. ^ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Cambridge University Press (1971), ISBN 0-521-07988-8 , §7, Theorem 2
5. ^ I. Vidav: A metrical characterization of the self adjoint operators , Mathematische Zeitschrift, Volume 66 (1956), pages 121-128
6. ^ FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862 , §38, Theorem 15
7. ↑ BJ Vowden: On the Gelfand-Naimark theorem , J. London Math Soc, Volume 42 (1967), pages 725-731..