The Banach–Lie algebra L(A) of multiplication operators on the JB∗-triple A is introduced and it is
shown that the hermitian part L(A)h of L(A) is a unital GM-space the base of the dual cone in the dual
GL-space (L(A)h)
∗ of which is affine isomorphic and weak∗-homeomorphic to the state space of L(A). In
the case in which A is a JBW∗-triple, it is shown that tripotents u and v in A are orthogonal if and only if
the corresponding multiplication operators in the unital GM-space L(A)h satisfy
0 D(u, u) +D(v, v) idA,

The hermitian part L(A)h of the Banach-Lie -algebra L(A) of multiplication operators
on the W-algebra A is a unital GM-space, the base of the dual cone in the dual GL-
space (L(A)h) of which is affine isomorphic and weak-homeomorphic to the state
space of L(A). It is shown that there exists a Lie -isomorphism from the quotient
(A 1 Aop)/K of an enveloping W-algebra A 1 Aop of A by a weak-closed Lie
-ideal K onto L(A), the restriction to the hermitian part ((A1Aop)/K)h of which is

The double backward method for nding zeros of the sum of two
maximal monotone operators is investigated. This method was initially intro-
duced by Passty in 1979 who used an equal index to the resolvents of both oper-
ators. In this paper, we use distinct indices in order to see different roles played
by the two operators in the double backward method. Under certain conditions
on the indices, we prove the ergodic convergence of our method.

The lasso of Tibshirani is a popular model for variable selections.
The elastic net of Zou and Hastie applies Tikhonov's regularization to the
lasso to break some limitations of the lasso in the case where the number of
predictors is much bigger than the number of observations, or where a group
of variables have pairwise high correlations. We generalize the elastic net by
replacing Tikhonov's regularization with a more general `p-norm regularization
which we refer to as the p-elastic net. One diculty for dealing with the p-