Dixmier's approximation theorem

The approximation of Dixmier , named after Jacques Dixmier is a set of the mathematical theory of von Neumann algebras . It says that one can approximate an element of the center by means of convex combinations of the unitary conjugates of an element of a Von Neumann algebra .

Formulation of the sentence

Let it be the group of unitary elements and the center of a Von Neumann algebra . Then applies to every element ${\ displaystyle U}$ ${\ displaystyle Z}$ ${\ displaystyle A}$ ${\ displaystyle a \ in A}$

{\ displaystyle {\ overline {\ mathrm {conv} (\ {uau ^ {*} | \, u \ in U \})}} \ cap Z \ not = \ emptyset}

.

The formation of the convex envelope and the horizontal line denote the standard closure . ${\ displaystyle \ mathrm {conv}}$

Additional: Is a finite von Neumann algebra , so the above average is a singleton. It consists of the image of the trace of . ${\ displaystyle A}$ ${\ displaystyle a}$

Applications

The fact that for finite Von Neumann algebras the intersection of the norm closure of the convex hull of the elements with the center is one-element can be used to show the existence of the trace. Each element of Von Neumann's algebra is mapped to the uniquely determined element of this average, which defines the trace. This procedure is detailed in the cited textbook by Kadison and Ringrose. ${\ displaystyle uau ^ {*}}$
With the help of Dixmier's approximation theorem, it can be shown that for a Von Neumann algebra with a center, the mapping ${\ displaystyle A}$ ${\ displaystyle Z}$

{\ displaystyle M \ mapsto Z \ cap M}

is a bijection of the set of all maximal , two-sided ideals of onto the set of maximal ideals of the center.

{\ displaystyle A}

Individual evidence

^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 8.3.5
^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter III.5.1, Theorem 1
^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 8.3.6
^ Jacques Dixmier: Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter III.5.2, Corollary 1

[1] RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 8.3.5

[2] Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter III.5.1, Theorem 1

[3] RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 8.3.6

[4] Jacques Dixmier: Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter III.5.2, Corollary 1