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Tracial State

A tracial state on a unital C*-algebra A is a positive linear functional tau:A->C satisfying tau(1)=1 and

tau(ab)=tau(ba)

for all a,b in A. For example, the normalized matrix trace tau(a)=n^(-1)Tr(a) is a tracial state on M_n(C). A tracial state is faithful when tau(a^*a)=0 implies a=0. A normalized 2-quasitrace has weaker additivity requirements; full linearity is the issue in Kaplansky's quasitrace conjecture.

See also

C*-Algebra, Extended Trace Cone, Free Entropy, Free Fisher Information, Free Independence, Kaplansky's Quasitrace Conjecture, Operator Algebra State, Quasitrace, Stably Finite C*-Algebra

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References

Anantharaman, C. and Popa, S. An Introduction to II_1 Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.

Cite this as:

Weisstein, Eric W. "Tracial State." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TracialState.html

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