During the study of quasidiagonal C$^\ast$-algebras, I came across stably finite ones (the former property implies the latter). When I looked up the definition of stably finite C$^\ast$-algebras I saw several version. I presume these are equivalent, at least to some extend. They are:

(1) A C$^\ast$-algebra $A$ is finite if all projetions are finite and $A$ admits an approximate unit of projections. As such, $A$ is stably finite provided $A\otimes \mathbb{K}$ is finite (compact operators on an infinite dimensional seaprable Hilbert space).

(2) A unital C$^\ast$-algebra is finite if $1_A$ is finite and a nonunital one is finite if its minimal unitization is finite. As such, we call $A$ stably finite provided $M_n(A)$ is finite for every $n\in \mathbb{N}$.

To me (1) is seemingly more general, although I doubt it. I am mostly confused about the stably finiteness condition. It seems that (2) should imply (1) by regarding the stabilization as the inductive limit over matrix algebra $M_n(A)$ with the canonical embeddings. Also, this seems odd since $A\otimes \mathbb{K}$ is not unital even if $A$ is.

I am probably just confused, so any comments or references etc. are greatly appreciated.


Indeed, (1) is a more general notion of finiteness. There are examples when $1_{A^\sim} \in A^\sim$ (the unitization of $A$) is finite, but not every projection in $A$ is finite. For simple C$^\ast$-algebras these notions are equivalent.

To see that $A \otimes \mathbb K$ is finite, you have to use that projections in $A \otimes \mathbb K$ can be approximated by projections in $\bigcup_{\mathbb N} M_n(A)$.

The details can be found in V.2.2.1 and V.2.2.2 of Blackadar's book "Theory of C$^\ast$-algebras and von Neumann algebras".

  • $\begingroup$ Lovely, thank you for the clarifying and the reference :) $\endgroup$ – Munk Dec 30 '16 at 12:50

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