This is an announcement for the paper "Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces" by Valentin Ferenczi and Eloi Medina Galego. Abstract: We improve the known results about the complexity of the relation of isomorphism between separable Banach spaces up to Borel reducibility, and we achieve this using the classical spaces $c_0$, $\ell_p$ and $L_p$, $1 \leq p <2$. More precisely, we show that the relation $E_{K_{\sigma}}$ is Borel reducible to isomorphism and complemented biembeddability between subspaces of $c_0$ or $\ell_p, 1 \leq p <2$. We show that the relation $E_{K_{\sigma}} \otimes =^+$ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz equivalence between subspaces of $L_p, 1 \leq p <2$. Archive classification: Functional Analysis; Logic Mathematics Subject Classification: 03E15; 46B03 Remarks: 22 pages; 2 figures The source file(s), sjm16.tex: 74499 bytes, is(are) stored in gzipped form as 0406477.gz with size 22kb. The corresponding postcript file has gzipped size 86kb. Submitted from: eloi@ime.usp.br The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0406477 or http://arXiv.org/abs/math.FA/0406477 or by email in unzipped form by transmitting an empty message with subject line uget 0406477 or in gzipped form by using subject line get 0406477 to: math@arXiv.org.