Atlas: A new look at biseparating maps I by Melvin Henriksen

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Special Session in Topological Methods
April 13-14, 1996
Courant Institute, New York University
New York, NY, USA

Organizers
L. Narici, E. Beckenstein, C. Traina

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A new look at biseparating maps I
by
Melvin Henriksen
Harvey Mudd College

Suppose A and B are commutative reduced unital algebras over a field F. A bijection h: A --> B is called a biseparating map if fg = 0 in A if and only if h(f)h(g) = 0 in B. An ideal I of A is called an F-ideal if there is an F-homomorphism of A onto F with kernel I, is called a d-ideal if it contains the annihilator of each of its elements and is called pseudoprime if fg = 0 implies one of f or g is in I.

Theorem If

the intersection of all the F-ideals of A and B is {0},

every pseudoprime d-ideal of A is prime

each proper prime ideal of A is contained in a unique maximal ideal, and h: A --> B is an F-linear biseparating map,

then the spaces Max(A) and Max(B) are homeomorphic in the hull-kernel topology, as are the subspaces F(A), F(B) of F-ideals of A and B, respectively, and h(a)(M) = M(a)h(1) for each M in F(A), where M(a) is the image in F of a mod M.

This generalizes a result of J. Araujo, E. Beckenstein and L. Narici who proved this in the case F is either the real or complex field and A = C(X, F), B = C(Y, F) are the algebras of all continuous F-valued functions on realcompact spaces X, Y. (See J. Math. Anal. Appl. 192, 258-265, (1995).) The proof techniques for the new results are algebraic in character.

Date received: December 30, 1995