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Hattori-Stallings Trace

The Hattori-Stallings trace over a unit ring R sends an idempotent matrix e in M_q(R) to its diagonal sum modulo additive commutators. Writing [R,R]_(add) for the additive subgroup generated by ab-ba, its value is

HS(e)=sum_(i=1)^qe_(ii) (mod[R,R]_(add)).

It depends only on the associated finitely generated projective module and extends additively to a map on K_0(R) in algebraic K-theory.

For the complex discrete group algebra l^1(G), its conjugacy-class coordinates are

tau_C(e)=sum_(g in C)sum_(i=1)^qe_(ii)(g).

The sums converge absolutely, and C ranges over conjugacy classes in G. The l1-Bass conjecture predicts finite support on classes of elements of finite order. Finite order refers to the elements, not to the cardinality of their conjugacy classes.

See also

Algebraic K-Theory, Conjugacy Class, Discrete Group Algebra, l1-Bass Conjecture, Projective Module

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References

Berrick, A. J.; Chatterji, I.; and Mislin, G. "From Acyclic Groups to the Bass Conjecture for Amenable Groups." Math. Ann. 329, 597-621, 2004. https://doi.org/10.1007/s00208-004-0521-6.

Cite this as:

Weisstein, Eric W. "Hattori-Stallings Trace." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hattori-StallingsTrace.html

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