SUPPOSE D IS AN INNER DERIVATION ON A BANACH ALGEBRA AINTO THE DUAL SPACE X*. SOME CONDITIONS UNDER WHICH THE SECOND DUAL OF D IS A DERIVATION, ARE INVESTIGATED. AMONG THE ARGUMENTS, THE REGULARITY OF BOUNDED BILINEAR MAPPINGS ARE ALSO DISCUSSED AND SOME OF THEM HAVE BEEN USED TO OBTAIN THE RESULTS.

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LET A BE A BANACH ALGEBRA AND X BE AN ARBITRARY BANACH A -BIMODULE. IN THIS PAPER, WE STUDY SECOND TRANSPOSE A DERIVATION WITH VALUE IN DUAL BANACH A -MODULE X*: INDEED, FOR A CONTINUOUS DERIVATION D: A®X* WE OBTAIN A NECESSARY AND SUFFICIENT CONDITION SUCH THAT THE BOUNDED LINEAR MAP Ù O D”: A ** ®X *** TO BE A DERIVATION, WHERE Ù IS COMPOSITION OF RESTRICTION AND CANONICAL INJECTION MAPS.

IN THIS TALK WE EXTEND THE TERNARY PRODUCT OF A C∗−TERNARYALGEBRA M TO SOME TERNARY PRODUCTS ON THE SECOND DUAL M∗∗ OF M. WE THEN SHOW THAT M∗∗ IS ALSO A C∗−TERNARY ALGEBRA.

IN THIS PAPER, WE SHOW THAT UNDER CERTAIN CONDITIONS, SEMISIMPLICITY OF THE BANACH ALGEBRA A IMPLIES THAT OF A”, WHERE A” IS THE SECOND DUAL OF A ENDOWED WITH EITHER ARENS PRODUCT.