Morita equivalence

[Urs Schreiber]

Let K be some field. Then the algebra M(n,K) of nxn matrices with
entries in that field is Morita equivalent to K, regarded as an algebra
over itself.

What happens for n --> infinity?

I expect that the algebra of compact operators K(H) on some seperable
Hilbert space H over K is Morita equivalent to K. But I am not sure.

In general, can one characterize all algebras over K that are Morita
equivalent to K?

[Mariano Suárez-Alvarez]

Urs Schreiber wrote:
> Let K be some field. Then the algebra M(n,K) of nxn matrices with
> entries in that field is Morita equivalent to K, regarded as an algebra
> over itself.
>
> What happens for n --> infinity?
>
> I expect that the algebra of compact operators K(H) on some seperable
> Hilbert space H over K is Morita equivalent to K. But I am not sure.

This is indeed so for K = Complex numbers (when you look at
the algebras as C^*-algebras, at least) . I don't have a reference
at hand, though.

> In general, can one characterize all algebras over K that are Morita
> equivalent to K?

A K-algebra A is Morita equivalent to K if there exists a finitely
projective generator P of the category of A-modules such that
End_A(P) = k --- this follows from Morita's theorem. I guess you
want more than this, though ;-)

-- m

[Aaron Bergman]

In article <ead71m$ete$1...@news.ks.uiuc.edu>,
"Mariano Suárez-Alvarez" <mariano.su...@gmail.com> wrote:

I seem to remember a theorem that for some restricted class of C^*
algebras, Morita equivalence is the same as stable equivalence, where
stable equivalence means that the algebras are the same upon tensoring
with the algebra of compact operators on the infinite dimensional
separable Hilbert space.

I'd guess that I saw it in Raeburn and Williams, _Morita Equivalence and
Continuous Trace C^* Algebras_, for a precise statement.

Aaron