Maximum Modulus Algebras and Analytic Varieties

Operator Algebras for Analytic Varieties

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictionsMV of the multiplier algebraM of Drury-Arveson space to a holomorphic subvariety V of the unit ball. We find thatMV is completely isometrically isomorphic toMW if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the i...

On the Structure of Maximum Modulus Algebras

i.e., if every fER attains its maximum modulus on C. As a matter of convenience, we shall, in this paper, abbreviate "maximum modulus algebra on K" to "ilf-algebra." Examples of Af-algebras which come to mind immediately are (a) the algebra ft consisting of all functions which are continuous on K and analytic in U, (b) any subalgebra of ft, (c) any algebra (B which is equivalent to an algebra o...

A Strong Maximum Modulus Theorem for Maximal Function Algebras

Analytic Varieties versus Integral Varieties of Lie Algebras of Vector Fields

We associate to any germ of an analytic variety a Lie algebra of tangent vector fields, the tangent algebra. Conversely, to any Lie algebra of vector fields an analytic germ can be associated, the integral variety. The paper investigates properties of this correspondence: The set of all tangent algebras is characterized in purely Lie algebra theoretic terms. And it is shown that the tangent alg...

Varieties of Algebras and their Congruence Varieties

1. Congruence lattices. G. Birkhoff and O. Frink noted that the congruence lattice Con04) of an algebra A (with operations of finite rank) is algebraic or compactly generated. The celebrated Grätzer-Schmidt theorem states that, conversely, every algebraic lattice is isomorphic to Con(A) for some algebra A. The importance of this is obvious, for it shows that unless something more is known about...