What are some examples of C*-algebras that do not admit a bounded tracial linear functional?

Question

Realizing a C*-algebra, say $A$, as a subalgebra of some $B(H)$ we could equip it with the trace it inherits from $B(H)$. But this is unbounded and not even defined on most elements. To my knowledge (correct if me I'm wrong) it is not even intrinsic to the C*-algebra and depends on what faithful representation $A \rightarrow B(H)$ we used. Anyway, what can be said about C*-algebras/von Neumann algebras admitting a tracial, bounded, linear functional? In the VN case, can something be said about its projections?

Answer 1

This is really broad: traces play a crucial role both in the theories of C$^*$-algebras and von Neumann algebras.

In the case of von Neumann algebras, having a faithful tracial state is equivalent with being finite: so only types $I_n$ with $n<\infty$ and II$_1$ can appear in the central decomposition. The identity is always finite, so all projections are finite. In the case of a II$_1$-factor, the trace determines the equivalence classes of projections under Murray-von Neumann equivalence; the values of traces of projections make precisely the interval $[0,1]$.

For C$^*$-algebras, the trace space is part of Elliott's invariant, so people have paid a lot of attention to traces over the last few decades.

To answer the question in the title, von Neumann algebras with no trace are those that have an infinite projections. So they have at least a summand/integrand of types I$_\infty$, II$_\infty$, or III.

For C$^*$-algebras something similar happens. A C$^*$-algebra with an infinite projection cannot have a faithful trace. That's the case for instance with the Cuntz algebras.