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Weakly analytic sets for function algebra is studied by Arenson in (Arenson). Here, we study the concept of weakly analytic sets for Cartesian product of function algebras. We express the weakly analytic sets for Cartesian product of function algebra in terms of that for factor algebras. 1 Weakly analytic sets The concept of weakly analytic sets for a function algebra was introduced by Arenson (Arenson). The weakly analytic property is stronger than antisymmetry. The decomposition given by maximal antisymmetric sets for the Cartesian product is obtained in (Mehta and Mehta). The weakly analytic sets induces a decomposition which is finer than the Bishop decomposition. Here, we have obtained weakly analytic sets for the Cartesian product of function algebras in terms of factor algebras. Let X be a compact, Hausdorff space and ( ) C X be the Banach algebra of complex valued continuous functions on X . A complex function algebra is a closed subalgebra of ( ) C X , which contains constants and separates the points of X . Examples 1.1 1. Trivially, ( ) C X , where X is a compact, Hausdorff space is a function algebra on X . 2. Let = { :| | 1} D z z   . Then, ( ) A D the set of all functions continuous on D and analytic in the interior of D is a function algebra on D , known as the Disc algebra. 3. Let 1 = { :| | 1} D z z   , 2 = { :| 2 | 1} D z z    and 1 2 = D D D  . Also, let Kalpa Publications in Computing Volume 2, 2017, Pages 130–134 ICRISET2017. International Conference on Research and Innovations in Science, Engineering &Technology. Selected Papers in Computing R. Buyya, R. Ranjan, S. Dutta Roy, M. Raval, M. Zaveri, H. Patel, A. Ganatra, D.G. Thakore, T.A. Desai, Z.H. Shah, N.M. Patel, M.E. Shimpi, R.B. Gandhi, J.M. Rathod, B.C. Goradiya, M.S. Holia and D.K. Patel (eds.), ICRISET2017 (Kalpa Publications in Computing, vol. 2), pp. 130–134 = { ( ) : isanalyticin } o A f C D f D  , where o D denote interior of D . Then, A is a function algebra on D . We start with some basic concepts related to the Cartesian product of function algebras. Let X and Y are two compact, Hausdorff topological spaces with topologies X  and Y  respectively. Then, the disjoint union X Y  is a compact, Hausdorff topological space with the sum topology  defined as = { : , } X Y G X Y G X G Y          (Shah). We denote this topological space by X Y  . Also, the mapping : ( ) ( ) ( ) T C X C Y C X Y    defined by (( , )) = T f g h where ( ), ; ( ) = ( ), f w if w X h w g w if w Y      is an algebra isomorphism from ( ) ( ) C X C Y  on to ( ) C X Y  , where ( ) ( ) C X C Y  is Cartesian product of ( ) C X and ( ) C Y defined as ( ) ( ) = {( , ) : ( ), ( )} C X C Y f g f C X g C Y    . If A is a function algebra on X , B is a function algebra on Y , then A B  is a function algebra on X Y  with maximum norm defined as = { , } h max f g for = ( , ) h f g A B   (Patel). Remark 1.2 For a closed subset E X Y   , we can easily verify that ( ) ( ) ( ) = E E X E Y A B A B     , where F A denotes the uniform closure of the restriction algebra |F A in ( ) C F for a closed subset F of X . Also, if E X  (respectively ) E Y  then ( )E E A B A   (respectively ( )E E A B B   ). We need the definition of peak sets and weak peak sets for a function algebra which are required for our main result. Now onwards, A and B will denote function algebras on X and Y respectively. Definition 1.3 A nonempty subset S of X is called a peak set for A if here exists f A  such that | = 1 S f and | ( ) |< 1 f y for all \ y X S  . The function f is called a peaking function on S . The set S is called a weak peak set for A if it is an intersection of some collection of peak sets. For peak sets we know that 1. If (respectively ) S T is a peak set for (respectively ) A B , then (respectively ) S T is a peak set for A B  . Hence, S T  is a peak set for A B  . 2. If S is a peak set for A B  , then S X  and S Y  are peak sets for A and B respectively. Above results also remain valid if we replace “peak set" by “weak peak set". Next, we recall the definition of weakly analytic sets. Definition 1.4 (Arenson) A closed subset E of X is called a weakly analytic set for A , if every peak set S for E A , either coincides with E or it is nowhere dense in E . Equivalently, E is a weakly analytic set for A if = E S T  , where S is a peak set for E A and T is a closed set in E , then either = S E or = T E . A is called weakly analytic algebra if X is a weakly analytic set for A . Following result is known for weakly analytic sets. Weakly analytic sets for the Cartesian... H. S. Mehta, R. D. Mehta and A. N. Roghelia

Weakly analytic sets for the Cartesian product of function algebra