UHF algebra

UHF algebras are examined in the mathematical sub-area of functional analysis. It is a class of C * algebras , which are also called Glimm algebras after their discoverer James Glimm . The UHF algebras are simple , that is, apart from 0 and themselves, they have no two-sided ideals , and they can be used to construct certain Von Neumann algebras .

construction

Denote the C * -algebra of complex - matrices . If it is a divisor of , then let the * - homomorphism , which maps a matrix from onto the matrix that consists of copies of the starting matrix along the diagonal, for example ${\displaystyle M_{n}}$ ${\displaystyle n\times n}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle \iota :M_{n}\rightarrow M_{m}}$ ${\displaystyle M_{n}}$ ${\displaystyle m\times m}$ ${\displaystyle m/n}$

${\displaystyle M_{2}\rightarrow M_{6},{\begin{pmatrix}x_{11}&x_{12}\\x_{21}&x_{22}\end{pmatrix}}\mapsto {\begin{pmatrix}x_{11}&x_{12}&0&0&0&0\\x_{21}&x_{22}&0&0&0&0\\0&0&x_{11}&x_{12}&0&0\\0&0&x_{21}&x_{22}&0&0\\0&0&0&0&x_{11}&x_{12}\\0&0&0&0&x_{21}&x_{22}\end{pmatrix}}}$.

This * homomorphism is injective and maps the one element onto the one element. Since injective * -homomorphisms between C * -algebras are automatically isometric , one can understand in this sense as a subalgebra of , and instead of we simply write . ${\displaystyle M_{n}}$ ${\displaystyle M_{m}}$ ${\displaystyle \iota }$ ${\displaystyle M_{n}\subset M_{m}}$

If now is a sequence of natural numbers , one obtains a chain of inclusions : ${\displaystyle {\vec {n}}=(n_{k})_{k\in \mathbb {N} }}$ ${\displaystyle \geq 2}$

${\displaystyle M_{n_{1}}\subset M_{n_{1}\cdot n_{2}}\subset M_{n_{1}\cdot n_{2}\cdot n_{3}}\subset \ldots }$.

On the union there is then a unique norm that continues each of the C * norms from and therefore has all the properties of a C * norm except for completeness . The completion is therefore a C * algebra called UHF algebra or Glimm algebra of rank . ${\displaystyle \bigcup _{k=1}^{\infty }M_{n_{1}\cdot \ldots \cdot n_{k}}}$ ${\displaystyle M_{n_{1}\cdot \ldots \cdot n_{k}}}$ ${\displaystyle {\vec {n}}}$

properties

Isomorphies

The UHF algebras depend of course on the defining sequence . For every prime number let the supremum of all , so that a divisor of , where runs towards infinity. In this way, the defining sequence is assigned the sequence which, in analogy to the prime factorization of natural numbers, is also written as and called a supernatural number , which of course is only to be understood symbolically; runs through all prime numbers. It applies ${\displaystyle {\vec {n}}=(n_{k})_{k\in \mathbb {N} }}$ ${\displaystyle p}$ ${\displaystyle \delta _{\vec {n}}(p)\in \mathbb {N} \cup \{\infty \}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle p^{n}}$ ${\displaystyle n_{1}\cdot \ldots \cdot n_{k}}$ ${\displaystyle k}$ ${\displaystyle {\vec {n}}}$ ${\displaystyle \delta _{\vec {n}}=(\delta _{\vec {n}}(p))_{p}}$ ${\displaystyle \prod _{p}p^{\delta _{\vec {n}}(p)}}$ ${\displaystyle p}$

• Two UHF algebras of rank or are isomorphic if and only if the assigned supernatural numbers are equal, that is, if for all prime numbers . ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {m}}}$ ${\displaystyle \delta _{\vec {n}}(p)=\delta _{\vec {m}}(p)}$ ${\displaystyle p}$

This sentence can already be found in. In particular, there are uncountably many pairwise non-isomorphic UHF algebras.

UHF algebras as AF algebras

According to the above construction, UHF algebras are special AF algebras ; the latter, however, were only introduced later. If the rank is UHF algebra, the corresponding Bratteli diagram is given by ${\displaystyle {\vec {n}}=(n_{k})_{k\in \mathbb {N} }}$

${\displaystyle {\begin{matrix}&\rightrightarrows &&\rightrightarrows &&\rightrightarrows &\\n_{1}&\vdots &n_{1}\cdot n_{2}&\vdots &n_{1}\cdot n_{2}\cdot n_{3}&\vdots &n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}\ldots \\&\underbrace {\rightarrow } _{n_{2}\,{\text{mal}}}&&\underbrace {\rightarrow } _{n_{3}\,{\text{mal}}}&&\underbrace {\rightarrow } _{n_{4}\,{\text{mal}}}\end{matrix}}}$.

One immediately reads that all UHF algebras are simple, but this can also be shown without using the Bratteli diagrams. UHF algebras are also classified as AF algebras by their ordered, scaled K ₀ group , which is isomorphic to

${\displaystyle \left\{{\frac {a}{b}};\,a,b\in \mathbb {Z} ,b\neq 0,b|n_{1}\cdot \ldots \cdot n_{k}{\text{ für ein }}k\in \mathbb {N} \right\}\subset \mathbb {Q} }$

with the scale given by [0,1].

Representations

UHF algebras are anti-criminal . Every irreducible representation is true and its image contains no other compact operator apart from 0 . UHF algebras have an uncountable number of pairwise non-equivalent, irreducible representations.

Construction of factors

Each UHF algebra has a unique track condition , that is a continuous linear functional with , and for all elements . The associated GNS construction provides a representation on a Hilbert space . It can be shown that the bicommutant of the picture is a type II ₁ factor . ${\displaystyle A}$ ${\displaystyle \tau }$ ${\displaystyle \tau (x^{*}x)\geq 0}$ ${\displaystyle \tau (1)=1}$ ${\displaystyle \tau (xy)=\tau (yx)}$ ${\displaystyle x,y\in A}$ ${\displaystyle \pi _{\tau }:A\rightarrow L(H)}$ ${\displaystyle H}$ ${\displaystyle \pi _{\tau }(A)^{''}\subset L(H)}$

Factors are called hyperfinite if they are generated as Von-Neumann algebras by an increasing sequence of finite-dimensional Unter-von-Neumann algebras. The name of the UHF algebras is derived from this, because these are in such hyperfinite factors, UHF stands for uniformly hyperfinite .

The CAR algebra , which is equal to the UHF algebra with the supernatural number , plays a special role . Representations of this algebra are constructed in, the images of which have type III factors as bicommutants. ${\displaystyle 2^{\infty }}$

Individual evidence

1. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Chapter 6.4.2
2. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 6.4.6
3. ^ J. Glimm: On a certain class of operator algebras , Transactions of the Amer. Math. Soc., Vol. 95 (1960), pp. 318-340
4. ^ KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 , proof of Corollary IV.5.8
5. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 6.5.7
6. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Corollary 6.4.4
7. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , III.7.4, Theorem 3
8. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , Theorem 6.5.15