Constant functional calculus

The continuous functional calculus is one of the most important foundations of the mathematical theory of C * algebras .

In advanced theory, the applications of this functional calculus are so self-evident that they are often not even mentioned. One can say without exaggeration that the continuous functional calculus, which is also in the basic sets of Gelfand-Neumark infected, the difference between C * -algebras and general Banach, where you only a holomorphic functional calculus has accounts.

motivation

If one wants to construct a functional calculus for continuous functions on the spectrum of a Banach algebra element , it makes sense to approximate the continuous functions by polynomials according to Weierstraß's approximation theorem, to insert the element into these polynomials and to show that this approximates an element in . To approximate continuous functions on, one needs polynomials in two variables, or what amounts to the same thing, polynomials in and , where denotes the complex conjugation . If you have such a polynomial and you put in place of , it is not initially clear what should be put in place of . Because an involution of the complex numbers, we consider Banach with an involution * and sets the place of . Since the polynomial ring is commutative, one has to restrict oneself to Banach algebra elements in order to obtain a homomorphism , such elements are called normal . If there is a sequence of polynomials that converges uniformly to a continuous function, then it must still be ensured that the sequence in tends towards a limit value which one could then name. A detailed analysis of this convergence problem shows that one has to fall back on C ^* -algebras . These considerations lead to the so-called continuous functional calculus. ${\displaystyle \sigma (a)}$ ${\displaystyle a\in A}$ ${\displaystyle A}$ ${\displaystyle \sigma (a)\subset \mathbb {C} }$ ${\displaystyle z}$ ${\displaystyle {\overline {z}}}$ ${\displaystyle {\overline {z}}}$ ${\displaystyle p(z,{\overline {z}})}$ ${\displaystyle a}$ ${\displaystyle z}$ ${\displaystyle {\overline {z}}}$ ${\displaystyle z\mapsto {\overline {z}}}$ ${\displaystyle a^{*}}$ ${\displaystyle {\overline {z}}}$ ${\displaystyle {\mathbb {C} }[z,{\overline {z}}]}$ ${\displaystyle {\mathbb {C} }[z,{\overline {z}}]\rightarrow A}$ ${\displaystyle a^{*}a=aa^{*}}$ ${\displaystyle (p_{n}(z,{\overline {z}}))_{n}}$ ${\displaystyle \sigma (a)}$ ${\displaystyle (p_{n}(a,a^{*}))_{n}}$ ${\displaystyle A}$ ${\displaystyle f(a)}$

The constant functional calculus

• Let be a normal element of C ^* -algebra with unity and let be the algebra of continuous functions on . Then there is exactly one * homomorphism with and . ${\displaystyle a}$ ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle {\mathcal {C}}(\sigma (a))}$ ${\displaystyle \sigma (a)}$ ${\displaystyle \Phi _{a}:{\mathcal {C}}(\sigma (a))\rightarrow A}$ ${\displaystyle \Phi _{a}(1)\,=\,e}$ ${\displaystyle \Phi _{a}(z)\,=\,a}$
• ${\displaystyle \Phi _{a}}$is an isometric isomorphism of the sub-C ^* algebra generated by. ${\displaystyle a}$

Usually one uses suggestive . Then one can prove the following: ${\displaystyle f(a)\,:=\,\Phi _{a}(f)}$

• There are formulas , for all . ${\displaystyle (f+g)(a)\,=\,f(a)+g(a)}$ ${\displaystyle (f\cdot g)(a)=f(a)\cdot g(a)}$ ${\displaystyle f,g\in {\mathcal {C}}(\sigma (a))}$
• Applies to everyone . ${\displaystyle f\in {\mathcal {C}}(\sigma (a))}$ ${\displaystyle f(a)^{*}={\overline {f}}(a)}$
• Are and , then applies . ${\displaystyle f\in {\mathcal {C}}(\sigma (a))}$ ${\displaystyle g\in {\mathcal {C}}(\sigma (f(a)))}$ ${\displaystyle (g\circ f)(a)=g(f(a))}$
• The spectral mapping principle applies : for everyone . ${\displaystyle \sigma (f(a))\,=\,f(\sigma (a))}$ ${\displaystyle f\in C(\sigma (a))}$

So one can imagine actually inserting the Banach algebra elements into continuous functions; the obvious algebraic operations behave as expected.

The requirement for a single element is not a major limitation. If necessary, one can adjoint a unity and work in the thus enlarged C * -algebra . If then and with , then and . ${\displaystyle A_{1}}$ ${\displaystyle a\in A}$ ${\displaystyle f\in {\mathcal {C}}(\sigma (a))}$ ${\displaystyle f(0)\,=\,0}$ ${\displaystyle 0\in \sigma (a)}$ ${\displaystyle f(a)\in A\subset A_{1}}$

Applications

The following applications are typical and very simple examples of the numerous applications of the continuous functional calculus in the theory of C ^* algebras:

root

Let be a normal element of a C ^* -algebra. Then are equivalent: ${\displaystyle a}$

• ${\displaystyle a}$is positive , i.e. H. . ${\displaystyle \sigma (a)\subset [0,\infty )}$
• There is a self adjoint element with . ${\displaystyle b}$ ${\displaystyle a\,=\,b^{2}}$

If it is positive, then the restriction of the root function to continuous, and one can form by means of functional calculus . Since only takes real values, is what follows and is evident . ${\displaystyle a}$ ${\displaystyle w\colon \mathbb {R} _{0}^{+}\to \mathbb {R} ,\ x\mapsto {\sqrt {x}}}$ ${\displaystyle \sigma (a)}$ ${\displaystyle b=w(a)\in A}$ ${\displaystyle w}$ ${\displaystyle w={\overline {w}}}$ ${\displaystyle b=b^{*}}$ ${\displaystyle b^{2}=(w(a))^{2}\,=\,(w^{2})(a)=z(a)=a}$

If the reverse is true with self-adjoint , then where , and follow from the spectral mapping theorem . ${\displaystyle a=b^{2}}$ ${\displaystyle b}$ ${\displaystyle a=b^{2}=b^{*}b={\overline {z}}(b)\cdot z(b)=({\overline {z}}\cdot z)(b)=|z|^{2}(b)=q(b)}$ ${\displaystyle q(z):=|z|^{2}}$ ${\displaystyle \sigma (a)=\sigma (q(b))=q(\sigma (b))\subset [0,\infty )}$

Unitary elements

• If a self-adjoint element of a C ^* -algebra is one element , then is unitary. ${\displaystyle a}$ ${\displaystyle e}$ ${\displaystyle u=e^{ia}}$

It is with , because since it is self adjoint, it follows , i.e. H. is a function on the spectrum of . It follows by means of functional calculus , i.e. H. is unitary. ${\displaystyle u=f(a)}$ ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} ,\ x\mapsto e^{ix}}$ ${\displaystyle a}$ ${\displaystyle \sigma (a)\subset \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle f\cdot {\overline {f}}={\overline {f}}\cdot f=1}$ ${\displaystyle uu^{*}=u^{*}u=e}$ ${\displaystyle u}$

swell

• J. Dixmier : Les C * -algèbres et leurs représentations . Gauthier-Villars, 1969.
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras. 1983, ISBN 0123933013 .
• M. Takesaki : Theory of Operator Algebras I . Springer, 1979, 2002.