H * algebra

An H * -algebra is a mathematical structure that is examined in the mathematical sub-area of functional analysis. It is an involutive Banach algebra , which is at the same time a Hilbert space , together with a condition that connects the involution with the Hilbert space structure. A structural theory analogous to Artin-Wedderburn's theorem is obtained .

Definition of the H * -algebra

An involutive -Banach algebra is called H * -algebra if the following applies: ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$

• There is a scalar product up , so that for everyone ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle A}$ ${\displaystyle \|x\|^{2}=\langle x,x\rangle }$ ${\displaystyle x\in A}$
• The following applies to all : and . ${\displaystyle a,x,y\in A}$ ${\displaystyle \langle ax,y\rangle =\langle x,a^{*}y\rangle }$ ${\displaystyle \langle xa,y\rangle =\langle x,ya^{*}\rangle }$

The involution is denoted by *. The first condition says that the Banach algebra with its Banach algebra is a Hilbert space. Each defines a linear operator via left multiplication and a linear operator via right multiplication . The second condition then says that (or ) is the Hilbert space adjoint to (or ), in formulas (or ), where the * on the right for the Hilbert space adjunction, i.e. for the involution of the C * -Algebra of the bounded linear operators on the Hilbert space . In this way the involution of the Banach algebra is related to the Hilbert space structure. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle L_{a}:A\rightarrow A,\,x\mapsto ax}$ ${\displaystyle R_{a}:A\rightarrow A,\,x\mapsto xa}$ ${\displaystyle L_{a*}}$ ${\displaystyle R_{a*}}$ ${\displaystyle L_{a}}$ ${\displaystyle R_{a}}$ ${\displaystyle L_{a^{*}}\,=\,(L_{a})^{*}}$ ${\displaystyle R_{a^{*}}\,=\,(R_{a})^{*}}$ ${\displaystyle B(A)}$ ${\displaystyle A}$

Examples

• The Hilbert-Schmidt class over a Hilbert space is an H * -algebra, where the scalar product is given by . ${\displaystyle A={\mathcal {S}}(H)}$ ${\displaystyle H}$ ${\displaystyle \langle x,y\rangle =\operatorname {Spur} (y^{*}x)}$
• Let be a compact group and the Hilbert space L ² (G) . With the convolution as multiplication and the involution defined by , it becomes an H * -algebra. ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle f^{*}(s):={\overline {f(s^{-1})}}}$ ${\displaystyle A}$
• Let be any non-empty set and be a real number. For and define ${\displaystyle J}$ ${\displaystyle A:=\{a:J^{2}\rightarrow \mathbb {C} ;\,\sum _{i,j\in J}|a(i,j)|^{2}<\infty \}}$ ${\displaystyle \alpha \geq 1}$ ${\displaystyle a,b\in A}$ ${\displaystyle \lambda \in \mathbb {C} }$

${\displaystyle {\begin{array}{rcl}(\lambda a)(i,j)&:=&\lambda a(i,j)\\(a+b)(i,j)&:=&a(i,j)+b(i,j)\\(ab)(i,j)&:=&\sum _{k\in J}a(i,k)b(k,j)\\(a^{*})(i,j)&:=&{\overline {a(j,i)}}\\\langle a,b\rangle &:=&\alpha \sum _{i,j\in J}a(i,j){\overline {b(i,j)}}\\\|a\|&:=&\langle a,a\rangle ^{1/2}\end{array}}}$.

With these definitions it becomes an H * -algebra, the so-called full matrix algebra . In the case , the full matrix algebra is isometrically isomorphic to the Hilbert-Schmidt class . ${\displaystyle A}$ ${\displaystyle \alpha =1}$ ${\displaystyle {\mathcal {S}}(\ell ^{2}(J))}$

• A continuous analog of the full matrix algebra is obtained as follows. Define for functions ${\displaystyle f,g\in A:=L^{2}([0,1]\times [0,1])}$

${\displaystyle {\begin{array}{rcl}(fg)(s,t)&:=&\int _{0}^{1}f(s,r)g(r,t){\mathrm {d} }r\\(f^{*})(s,t)&:=&{\overline {f(t,s)}}\end{array}}}$.

With these definitions the Hilbert space becomes an H * -algebra. ${\displaystyle A}$

• The sequence space is a commutative H * -algebra with the component-wise explained multiplication and the involution defined by the component-wise complex conjugation . ${\displaystyle \ell ^{2}}$

Structure theory

The structural theory of H * -algebras, which is analogous to the Artin-Wedderburn theorem, was discovered in 1945 by Warren Ambrose .

1. Structure theorem

An H * -algebra breaks down into an orthogonal sum . It is the Jacobson radical of , and the completion of all finite sums of products of two elements is a semisimple H * algebra, that is its Jacobson radical is . ${\displaystyle A}$ ${\displaystyle A={\overline {A^{2}}}\oplus (A^{2})^{\perp }}$ ${\displaystyle (A^{2})^{\perp }=\{x\in A;\,xA=\{0\}\}=\{x\in A;\,Ax=\{0\}\}}$ ${\displaystyle A}$ ${\displaystyle {\overline {A^{2}}}}$ ${\displaystyle A}$ ${\displaystyle \{0\}}$

The product of two elements of the radical is 0. Therefore only the structure of semi-simple H * -algebras has to be investigated.

2. Structure theorem

A semi-simple H * -algebra breaks down into the orthogonal sum of the minimal, closed, two-sided ideals and thus into a direct sum of simple H * -algebras.

An H * -algebra is called simple if it has no non-trivial, two-sided, closed ideals. With that only the structure of simple H * -algebras has to be examined.

3. Structure theorem

A simple H * algebra is isometrically isomorphic to a full matrix algebra.

This reveals the structure of the H * -algebras: An H * -algebra is isometrically isomorphic to an orthogonal sum of a Hilbert space with zero multiplication and full matrix algebras. The Hilbert space with zero multiplication is the Jacobson radical. The individual summands of the direct sum can be the null space , they are then omitted.

See also

• Hilbert algebra

swell

• FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3540063862
• Warren Ambrose: Structure Theorems for a special class of Banach-Algebras , Trans. Amer. Math. Soc. 57 (1945), pp. 364-386