Let A be a finitedimensional
associative algebra with identity over a field k, M an Amodule which is
finitedimentional as a vector space over k, and E = Hom_{k}(M,M) the algebra of
linear transformations on M. For a ∈ A. Let a_{L} denote the linear transformation of
M given by a_{L}(x) = ax, for x ∈ M. Define the following subalgebras of
E:
A_{L}  = {a_{L} : a ∈ A}  
 C  = {f ∈ E : f(ax) = af(x) for each a ∈ A,x ∈ M}  
 D  = {f ∈ E : f(g(x)) = g(f(x)) for each g ∈ C,x ∈ M}.   
Clearly, A_{L} ⊆ D. Require M to be faithful. Then A is isomorphic to, and will be
identified with, A_{L}. If A = D, it is said that the pair (A,M) has the double
centralizer property.
A is called a QF1 algebra if (A,M) has the double centralizer property for each
faithful Amodule M.
The following results in the theory of QF1 algebras are obtained:
1. Let A be a commutative algebra over an arbitrary field. Then A is QF1 if and
only if A is Frobenius.
2. Let A be an algebra such that the simple left Amodules are onedimensional.
Suppose there exist distinct simple twosided ideals A_{1} and A_{2} contained in the
radical of A, and primitive idempotents e and f, such that eA_{k}f≠0, for k = 1,2.
Then A is not QF1.
3. Let A be an algebra with the properties that the simple left Amodules are
onedimensional, and the twosided ideal lattice of A is distributive. Then if A
satisfies any one of the following conditions, it is not QF1.
(a) There exist, for r ≧ 2, 2r distinct simple twosided ideals A_{uv} contained in the
radical, and primitive idempotents e_{iu} and e_{jv} for 1 ≦ u, v ≦ r, satisfying
e_{iu}A_{uv}E_{jv}≠0, where the index pair (u,v) ranges over the set
(b) There exist, for r ≧ 1, 2r + 2 distinct simple twosided ideals A_{uv} and A_{v}^{ρ}, for
(u,v) = (1,1),(1,2),⋯,(r − 1,r − 1),(r − 1,r), and (ρ,v) = (1,1),(2,1),(3,r), and
(4,r), and primitive idempotents e_{iu}, e_{jv}, and e_{kρ} satisfying e_{iu}A_{uv}e_{j}≠0 and
e_{kρ}A_{v}^{ρ}e_{jv}≠0, where (u,v) and (ρ,v) range over the index pairs indicated
above.
