ICC group

ICC groups in mathematics studied groups with infinite conjugacy . The abbreviation ICC stands for the English term infinite conjugacy classes .

definition

A group with at least two elements is called an ICC group if each of the different conjugation classes is infinite , with the neutral element of the group being. ${\ displaystyle \ {e \}}$ ${\ displaystyle e}$

This means that for each element, the set is an infinite set. ${\ displaystyle x \ not = e}$ ${\ displaystyle \ {yxy ^ {- 1} | \, y \ in G \}}$

Remarks

ICC groups are infinite and highly non-commutative. The center of an ICC group consists only of the neutral element.
The left regular representation of a discrete ICC group to generate a type II ₁ factor on . Hence their importance. ${\ displaystyle \ ell ^ {2} (G)}$

Examples

The free group with many producers is an ICC group.

{\ displaystyle n \ geq 2}

The group of finite permutations of is an ICC group; it is the group of the subgroup of the full permutation group generated by all transpositions on . ${\ displaystyle \ mathbb {N}}$ ${\ displaystyle \ mathbb {N}}$

Cartesian products of a finite number of ICC groups are ICC groups again.

Individual evidence

^ ^A ^b Masamichi Takesaki : Theory of Operator Algebra (= Encyclopaedia of mathematical sciences. Vol. 124 = Encyclopaedia of Mathematical Sciences. Operator Algebras and non-commutative Geometry. Vol. 5). Volume 1. 2nd printing of the 1st edition 1979. Springer, New York et al. 2002, ISBN 3-540-42248-X , chap. V, definition 7.10.
^ Li Bing-Ren: Introduction to Operator Algebras. World Scientific Pub. Co., Singapore et al. 1992, ISBN 981-02-0941-X , definition 7.3.19.
↑ Richard V. Kadison , John R. Ringrose (ed.): Fundamentals of the Theory of Operator Algebras (= Pure and Applied Mathematics. Vol. 100, Part 2). Academic Press, New York NY 1983, ISBN 0-1239-3302-1 , Theorem 6.7.5.
↑ Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras (= Pure and Applied Mathematics. Vol. 100, Part 2). Academic Press, New York NY 1983, ISBN 0-1239-3302-1 , Example 6.7.6
↑ Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras (= Pure and Applied Mathematics. Vol. 100, Part 2). Academic Press, New York NY 1983, ISBN 0-1239-3302-1 , Example 6.7.7

[Takesaki-1] A ^b Masamichi Takesaki : Theory of Operator Algebra (= Encyclopaedia of mathematical sciences. Vol. 124 = Encyclopaedia of Mathematical Sciences. Operator Algebras and non-commutative Geometry. Vol. 5). Volume 1. 2nd printing of the 1st edition 1979. Springer, New York et al. 2002, ISBN 3-540-42248-X , chap. V, definition 7.10.

[2] Li Bing-Ren: Introduction to Operator Algebras. World Scientific Pub. Co., Singapore et al. 1992, ISBN 981-02-0941-X , definition 7.3.19.

[3] Richard V. Kadison , John R. Ringrose (ed.): Fundamentals of the Theory of Operator Algebras (= Pure and Applied Mathematics. Vol. 100, Part 2). Academic Press, New York NY 1983, ISBN 0-1239-3302-1 , Theorem 6.7.5.

[4] Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras (= Pure and Applied Mathematics. Vol. 100, Part 2). Academic Press, New York NY 1983, ISBN 0-1239-3302-1 , Example 6.7.6

[5] Richard V. Kadison, John R. Ringrose: Fundamentals of the Theory of Operator Algebras (= Pure and Applied Mathematics. Vol. 100, Part 2). Academic Press, New York NY 1983, ISBN 0-1239-3302-1 , Example 6.7.7