Let $(X,{\cal S},\mu)$ be a probability space, and ${\cal T}$ a $\sigma$-subalgebra of ${\cal S}$. The conditional expectation operator $E^{\cal T}:L^1(X,{\cal S},\mu)\longrightarrow L^1(X,{\cal T},\mu)$ is determined by the relation $\int_T E^{\cal T}(f)d\mu=\int_T fd\mu$ for $E\in {\cal T}$, and all $f\in L^1(X,{\cal S},\mu)$.Existence and uniquencess of $E^{\cal T}$ follow from the Radon-Nikodym theorem. In particular, uniquencess shows that $E\in {\cal T}$ is idempotent, and since $||E\in {\cal T}||=1$, $E\in {\cal T}$ is acontractive projection. An easy argument withe simple functions also shows the fundamental relation $$E^{\cal T}(fg)=g.{\cal T}(f) \ \ \ (f\in L^1(X,{\cal S},\mu) \ , \ g \in L^\infty(X,{\cal T},\mu)).$$ lightly more generally, suppose, in the above situation that $S\in {\cal T}$ is fixed, and $k\geq 0$ is S-measurable and satisfies $\int_T kd\mu=\int_T {\cal X}d\mu $ for $T\in{\cal T}$, so that $E^{\cal T}k={\cal X}S$. Then again $f\mapsto k$. $E^{\cal T}(f)$ i a contractive projection on $L^\infty(X,{\cal S},\mu)$\\ A. Grothendieck [15] haz proved that if a projection in an $L_1$-spase is is contractive, i.e. it has norm$\leqq1$, then it range is isometric to a $L_1$-spase. Later, under the assumption that $({\cal X},{\cal S},\mu)$ is a finite measure spase, R.G. Douglas [9] has given a complete characterizatio of contractive projectio in $L_1(\omega,\sigma,m)$ related closely to the notion of conditional expectation. In fact the generat form of a contractive projection on $L_1(X,S,\mu)$ is $U_{\bar{\varphi}}(k.E^{\cal T})U_\varphi+V$ for some unimodular S-measurable $\varphi$, where $U_\varphi(f)=\varphi f$ and $V$ is a related contraction with $V^2=0$.\\ Douglas results have been extended by T. Ando [1] for $L_p$- space; $1 lt;+\infty$; he has proven that a contractive projection in a $L_p$- space; $1 lt;\infty$; over a finite measure space is similar to a conditional expectatio and hence, its range is isometric to a $L_p$- space, Obviously, the rage of a contractive projection in a separable $L_p$- space has the same structure.\\ Contractive projections have been studied in other situations, see, for example, [1,14,16,33,41]. In many cases it is known that the rage of such projection on certin algebras is often imbued with algebraic truture of its own, and the projection itself is a cinditionl expectation operator. \\ For example, see [17]. The current paper has the same flavour, though from the different perspective of hypothesising algebric conditions o the range. some other recent papers concerned with the conditional expectation roperty in related setting are [11,13,34].\\ In this Thesis we shall study contractive projections whose range is a subalgebra on vaious classes of Bannach algebras, particularly those associated with locally compact groups.